Critical Ising model on random triangulations of the disk: enumeration and local limits

We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte's equation, extending a previous result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of finite clusters are also obtained.


Introduction
Recent years have seen an increasing number of works devoted to random planar maps decorated by additional combinatorial structures such as trees, orientations and spin models. We refer to [11] for a survey from an enumerative combinatorics point of view. From a probabilistic point of view, one important motivation for studying decorated random maps is to understand models of two-dimensional random geometry that escape from the now well-understood universality class of the Brownian map [26,27]. This is in turn motivated by an effort to give a solid mathematical foundation to the physical theory of Liouville quantum gravity by discretization [2]. The critical Ising model is one of the simplest combinatorial structures that, when coupled to a random planar map, have a non-trivial impact on the geometry of the latter. The systematic study of the Ising model on random lattices was pioneered by Boulatov and Kazakov back in the eighties [23,10]. Using relations to the two-matrix model, they computed the partition function of the Ising model on random triangulations and quadrangulations in the thermodynamic limit, identifying its phase transitions and computing the associated critical exponents. This approach was later refined and generalized to deal with Ising models on more general maps as well as the Potts model [19,18]. A more mathematical derivation of the partition function on the discrete level was later given by Bernardi, Bousquet-Mélou and Schaeffer in [13,7]. In these works, the partition function is shown to be algebraic and having a rational parametrization. Our work complements the ones in [10,7] by dealing with Ising-decorated triangulations with a large boundary and a Dobrushin boundary condition. In addition, we exploit these combinatorial results using the so-called peeling process to derive some scaling limits of quantities describing the geometry of the Ising-interface, and ultimately the local limit of the Ising-decorated random maps themselves.
Let us define our conventions and terminology before stating the main results.
Planar maps. We refer to [28,16] for self-contained introductions to random planar maps.
Here we consider planar maps in which loops and multiple edges are allowed. A map is rooted when it has a distinguished corner. This corner determines a distinguished vertex ρ, called In the following, all maps are assumed to be planar and rooted.
A map is a triangulation of the -gon ( ≥ 1) if the internal faces all have degree three, and the contour of its external face is a simple closed path (i.e. it visits each vertex at most once) of length . The number is called the perimeter of the triangulation, and an edge (resp. vertex) adjacent to the external face is called a boundary edge (resp. boundary vertex). Figure 1(a) gives an example of a triangulation of the 7-gon. By convention, the edge map -the map containing only one edge and no internal face -is a triangulation of the 2-gon.
Bicolored triangulations of the (p, q)-gon. We consider the Ising model with spins on the internal faces of a triangulation of polygon. The triangulation together with an Ising spin configuration on it is represented by a couple (t, σ) where σ ∈ {+, -} F (t) . An edge e of t is said to be monochromatic if the spins on both sides of e are the same. When e is a boundary edge, this definition requires a boundary condition which specifies a spin outside each boundary edge. By an abuse of notation, we consider the information about the boundary condition to be contained in the coloring σ, and denote by E(t, σ) the set of monochromatic edges in (t, σ).
In this work, we concentrate on the Dobrushin boundary conditions which assign a sequence of spins of the form + · · · + -· · ·to the boundary edges in the counter-clockwise order starting from the origin.
Let p and q be respectively the numbers of + and of -in this sequence. Then we call (t, σ) a bicolored triangulation of the (p, q)-gon. Figure 1(b) gives an example in the case p = 3 and q = 4. We denote by BT p,q the set of all bicolored triangulation of the (p, q)-gon.
where ν > 0 is related to the coupling constant of the Ising model, and t is a parameter that controls the volume of the triangulation. When q = 0 and p is small, the above generating function has already been computed by Bernardi and Bousquet-Mélou in [7]. (More precisely, they computed the generating function of a model that is dual to ours. See Section 3.2 for more details.) A part of their result can be translated in our setting as follows.
We enumerate the elements of BT p,q by the generating function z p,q (ν, t) = (t,σ)∈BT p,q ν |E(t,σ)| t |F (t)| where ν > 0 is the exponential of the inverse temperature for the Ising model, and t is a parameter that controls the volume of the triangulation. When q = 0 and p is small, the above generating function has already been computed by Bernardi and Bousquet-Mélou in [7]. (More precisely, they computed the generating function of a model that is dual to ours. See Section 3.2 for more details.) A part of their result can be translated in our setting as follows.
This result suggests that ν c = 1 + 2 √ 7 is the unique value of ν at which the asymptotic behavior of the Ising-decorated random triangulation escapes from the universality class of the Brownian map (corresponding to ν = 1). The asymptotic form is also in agreement with the relation of Knizhnik, Polyakov and Zamolodchikov between the string susceptibility exponent γ and the central charge c of a CFT on a surface of genus zero, given in ( [24]): it can be written as [t n ]z 1,0 (ν, t) ∼ n→∞ κ(ν)τ (ν) −n n γ(ν)−2 , where the string susceptibility exponent is γ c := γ(ν c ) = − 1 3 and γ(ν) = − 1 2 otherwise, corresponding to a CFT with central charge c = 1 2 and c = 0, respectively. See Formula (4.223) in [2]. In the sense of CFT, the former corresponds to the critical Ising model whereas the latter to pure gravity.
In this work, we will concentrate on the critical value of the parameters, and leave the general case, as well as the phase transitions, to an upcoming work. In all that follows, we fix (ν, t) = (ν c , t c ) and write z p,q = z p,q (ν c , t c ).
Theorem 1 (Asymptotics of z p,q ). The generating function Z(u, v) = p,q≥0 z p,q u p v q is algebraic and can be expressed in terms of a rational parametrization which is described in Section 3.3 and given explicitly in [1]. The asymptotics of the coefficients z p,q are given by where u c = 6 5 (7 + √ 7)t c and b = − 27 20 ( 3 2 ) 2/3 , and the sequence (a p ) p≥0 is determined by its generating function A(u) = p≥0 a p u p given by the following rational parametrization: Under P p,q , the law of the spin configuration σ conditionally on t is given by the classical Ising model on t. And when ν = 1, the triangulation t follows the distribution of a Boltzmann triangulation of the (p + q)-gon as introduced in [5], with a weight 2t c per internal face. For these reasons we call P p,q the law of a (critical) Boltzmann Ising-triangulation of the (p, q)-gon. The expectation associated to P p,q is denoted E p,q . In order to extract information on the geometry of Boltzmann Ising-triangulations from Theorem 1, we use a peeling process that explores the triangulation along the Ising-interface. 1 More precisely, an interface refers to a non-self-intersecting (but not necessarily simple) path formed by non-monochromatic edges. Assuming that the boundary of (t, σ) is not monochromatic, there must be exactly two boundary vertices where the + and -boundary components meet. One of them is the origin ρ. We call ρ † the other one. We denote by I the left-most interface from ρ to ρ † as given in Figure 2(a). 2 We will consider a peeling process that explores I by revealing one triangle adjacent to I at each step, and possibly swallowing a finite number of other triangles. Formally, we define the peeling process as an increasing sequence of explored maps (e n ) n≥0 . The precise definition of e n will be left to Section 2.2. See Figure 2  Notice that the perimeter variations (X n , Y n ) can be read from the map e n as X n = E + n − S + n and Y n = En − Sn .
The peeling process can also be encoded by a sequence of peeling events (S n ) n≥1 taking values in some countable set of symbols, where S n indicates the position of the triangle revealed at time n relative to the explored map e n−1 . The detailed definition is again left to Section 2.2. The sequence (S n ) n≥1 contains slightly less information than (e n ) n≥0 , but it has the advantage that its law can be written down fairly easily and one can perform explicit computations with it. We denote by P p,q the law of the sequence (S n ) n≥1 under P p,q .
In order to understand the geometry of large Boltzmann Ising-triangulations, we want to study the peeling process in the limit p, q → ∞. The regime where p and q go to infinity at comparable speeds is probably the most natural and interesting one. But currently we do not know how to extract the asymptotics of z p,q from its generating function in this regime. Instead, we will look into the regime where q goes to infinity before p. The first step consists of showing that the law of the sequence P p,q converges weakly as follows: Proposition 2. P p,q − −− → q→∞ P p − −− → p→∞ P ∞ , where P p and P ∞ are probability distributions.
The distributions P p and P ∞ will be constructed explicitly in Section 4.1, thus no tightness argument is needed in the proof of the above convergence. Geometrically, Proposition 2 should be understood as the convergence in distribution of the explored map e n for any fixed n.
The perimeter processes and their scaling limits. One crucial point in the definition of the peeling process is that the unexplored map, i.e. the complement of the explored map e n , remains an Ising-triangulation with Dobrushin boundary condition for all n. We denote by (P n , Q n ) the boundary condition of the unexplored map at time n, and by (X n , Y n ) its variations, that is, X n = P n − P 0 and Y n = Q n − Q 0 . Geometrically, X n (resp. Y n ) is the number of newly discovered + boundary edges (resp. -boundary edges), minus the number of + boundary edges (resp. -boundary edges) swallowed by the peeling process up to time n. See Figure 2(b). It will be clear from the definition of the peeling process that (X n , Y n ) is a deterministic function of the peeling events (S k ) 1≤k≤n , with a functional relation independent of the initial condition (p, q). This allows us to define the law of the process (X n , Y n ) n≥0 under P ∞ despite the fact that P n = Q n = ∞ almost surely in this case. Similarly, (X n , Y n ) n≥0 is also welldefined under P p . However, it is easier to study the process (P n ) n≥0 in this case because it is Markovian under P p . These processes have the following scaling limits.
Theorem 3 (Scaling limit of the perimeter processes).
(1) Under P ∞ , the process (X n , Y n ) n≥0 is a random walk on Z 2 starting from (0, 0). Its two components have the same positive drift: Moreover, the fluctuation of (X n , Y n ) n≥0 around its mean has the following scaling limit: where X and Y are two independent spectrally-negative 4 3 -stable Lévy processes of Lévy measure cx |x| 7/3 1 {x<0} dx and cy |y| 7/3 1 {y<0} dy, for some explicit constants c x > c y > 0. (2) Under P p , the process (P n ) n≥0 is a Markov chain on Z ≥0 which starts from p and hits zero almost surely in finite time. It has the following scaling limit: where (D t ) t≥0 is the deterministic drift process (1 + µt) t≥0 that jumps to zero and stays there after a random time ζ whose law is given by Both convergences take place in distribution with respect to the Skorokhod topology.
An important point in Theorem 3(1) is that µ, the common drift of (X n ) n≥0 and (Y n ) n≥0 , is strictly positive so that both X n and Y n tend to +∞ when n → ∞. Geometrically, it means that under P ∞ , the peeling process discovers more and more edges on both sides of the interface I and comes back to the boundary only finitely many times. This is in contrast with the behavior of the percolation interface on uniform random maps of the half plane (e.g. the UIHPT) with the same boundary condition, which comes back to the boundary infinitely often (see [3,4]). This difference of the interface behaviour is reminiscent to the difference of SLE(3) and SLE (6), which arise respectively as scaling limits of critical Ising and percolation interfaces on regular lattices [14,32].
Theorem 3(2) says that on time scales n p, the process (X n ) n≥0 under P p increases with a drift µ like under P ∞ . However on the time scale n = O(p), the effect of the finiteness of the + boundary appears and makes P n hit zero in finite time. Geometrically, the large negative jump of (P n ) n≥0 corresponds to the first time that the peeling process hits a boundary vertex close to ρ † , swallowing most of the + edges on the boundary. The random time ζ should be interpreted as a length: for large p, the total length of the interface I under P p is almost surely finite and roughly ζp. More discussion on ζ is given in Section 6.
Notice that in Theorem 3(1), although the drifts are equal, there is an asymmetry between the fluctuations of the processes (X n ) n≥0 and (Y n ) n≥0 . This is not surprising because they are defined by the peeling process that explores the left-most interface. Nevertheless, this asymmetry is not related to the fact that we have taken first the limit q → ∞ and then the limit p → ∞. In fact, one can check that taking the limit p → ∞ and then q → ∞ yields the same distribution P ∞ . See the discussion on the peeling process along the right-most interface in Section 6. We conjecture that the distribution P ∞ actually arises when p, q → ∞ at any relative speed.
Local limits and geometry. Another way to improve Proposition 2 is to strengthen it to the local convergence of the underlying map. The local distance between bicolored maps is a straightforward generalization of local distance between uncolored maps: and [t, σ] r denotes the ball of radius r around the origin in (t, σ) which takes into account the colors of the faces. See Section 5.2 for a more precise definition of [t, σ] r . Similarly to the uncolored maps, the set BT of (finite) bicolored triangulations of polygon is a metric space under d loc . Let BT be its Cauchy completion.
Recall that an (infinite) graph is one-ended if the complement of any finite subgraph has exactly one infinite connected component. It is well known that a one-ended map has either zero or one face of infinite degree [16]. We call an element of BT \ BT a bicolored triangulation of the half plane if it is one-ended and its external face has infinite degree. A such triangulation has a proper embedding in the upper half plane without accumulation point and such that the boundary coincides with the real axis, hence the name. We denote by BT ∞ the set of all bicolored triangulations of the half plane.

Theorem 4 (Local limits of Ising-triangulation).
(1) There exist probability distributions P p and P ∞ supported on BT ∞ , such that weakly. In addition, if P p,(q 1 ,q 2 ) denotes the pushforward of P p,q 1 +q 2 by the mapping that translates the origin q 1 edges to the left along the boundary, then for all fixed p ≥ 0, we have P p,(q 1 ,q 2 ) dloc − − → P 0 weakly as q 1 , q 2 → ∞.
(2) P p -almost surely, (t, σ) contains only one infinite spin cluster, which is of spin -.
(3) P ∞ -almost surely, (t, σ) contains exactly two infinite spin clusters. One of them is of spin + on the right of the root, and the other is of spinon the left of the root. They are separated by a strip of finite clusters, which only touches the boundary of (t, σ) in a finite interval.
See Figure 3 for an illustration of the cluster structure in the Ising-triangulations of laws P p and P ∞ .
The construction of the limits P p and P ∞ is based on the laws P p and P ∞ of the peeling process in Proposition 2. Under P p , one can extend the peeling process after it finishes exploring the left-most interface I, in such a way that the explored map e n eventually covers all the internal faces of the Ising-triangulation. Consequently P p can be constructed directly as the law of the union ∪ n≥0 e n under P p . However, almost surely under P ∞ , the interface I is infinite and visits the boundary of (t, σ) only finitely many times (see the discussion after Theorem 3). Thus the peeling process only explores the faces of (t, σ) along a strip Figure 3: An artistic representation of the cluster structure of an Ising-triangulation of distribution P p and P ∞ . The dashed lines in (b) highlight the strip of finite clusters separating the two infinite clusters.
around the interface I. For this reason, the Ising-triangulation of law P ∞ is constructed by gluing two infinite bicolored triangulations to both sides of the strip given by ∪ n≥0 e n under P ∞ . The proof of the convergences in Theorem 4(1) follows closely the above construction of the distributions P p and P ∞ . The structure of the proof is summarized in Figure 7 at the beginning of Section 5. The statements (2) and (3) of Theorem 4 are direct consequences of our construction of the distributions P p and P ∞ . More discussions about them, as well as about other properties of the spin clusters under P p and P ∞ , will be given in Section 6. The rest of the paper is organized as follows.
We derive the so-called Tutte's equation (or loop equation) satisfied by Z(u, v) in Section 2.1 and define the peeling process of a bicolored triangulations of the (p, q)-gon in Section 2.2. The derivation is formulated in probabilistic language to highlight its relation with the first step of the peeling process.
For our model, Tutte's equation is a functional equation with two catalytic variables. In Section 3.1 we eliminate one of the catalytic variables by coefficient extractions, leading to a functional equation with one catalytic variable for Z(u, 0). Section 3.2 details the connection between our model and a model studied in [7], which is then used to translate some of their results (in particular Proposition A) in our setting. These results can also be obtained independently via a trick due to Tutte, which is presented in the Appendix A. Section 3.3 solves the functional equation on Z(u, v) at the critical point (ν, t) = (ν c , t c ) by a rational parametrization, and completes the proof of Theorem 1 with standard methods of singularity analysis. Some specific techniques for conducting singularity analysis using rational parametrizations are summarized in Appendix B.
Section 4 is devoted to the study of the limits of the peeling process and the associated perimeter processes, and the proof of Theorem 3. It also includes an important one-jump lemma on the perimeter processes, which is proven in Appendix C. In Section 5 we construct the distributions P p and P ∞ and prove the local convergences in Theorem 4(1). Finally, we discuss in Section 6 some properties of the spins clusters and the interfaces that follows from our construction of the infinite Ising-triangulation of law P p and P ∞ . It contains the proof of Theorem 4(2-3) and a scaling limit result for the perimeter of a spin cluster.
Acknowledgments. The authors thank M. Bousquet-Mélou, J. Bouttier, N. Curien, B. Eynard, K. Izyurov and A. Kupiainen for enlightening discussions and guidance. L. Chen acknowledges the support from the Agence National de la Recherche via Grant ANR-14-CE25-0014 (ANR GRAAL) and ANR-12-JS02-0001 (ANR CARTAPLUS). Both authors have been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project No. 271983), and by the ERC Advanced Grant 741487 (QFPROBA).

Tutte's equation and peeling along the interface
Recall that we have fixed the critical parameters (ν c , t c ) and defined Z(u, v) = p,q≥0 z p,q u p v q with z 0,0 = 1 and z p,q = z p,q (ν c , t c ) for p + q ≥ 1. However, many of the discussions below will be valid for any ν, t > 0 such that z p,q (ν, t) < ∞. In this case we will write (ν, t) instead of (ν c , t c ).
The primary goal of this section is to derive a recurrence relation for the double sequence (z p,q ) p,q≥0 , and then a functional equation -the so-called Tutte's equation (a.k.a. loop equation, or Schwinger-Dyson equation) -for its generating function. The basic idea, which goes back to Tutte [33], is to consider the removal of one face on the boundary, which relates one bicolored triangulation of polygon to other ones with fewer faces. We will present a probabilistic derivation of Tutte's equation. This is a bit more cumbersome than a direct combinatorial derivation, but will shed light on the relation between Tutte's equation and the peeling process, which we define in the second half of this section.

Derivation of Tutte's equation
Let p, q ≥ 0 so that the bicolored triangulation (t, σ) ∈ BT p,q+1 has at least one boundary edge with spin -. We remove the boundary edge e immediately on the left of the origin (which has spin -) and reveal the internal face f adjacent to it. It is possible that f does not exist if (p, q + 1) = (1, 1) or (0, 2). In this case t is the edge map and (t, σ) has a weight 1 or ν. When f does exist, let * ∈ {+, -} be the spin on f and v be the vertex at the corner of f not adjacent to e. There are three possibilities for the position of v. Figure 4: A graphical representation of the derivation of Tutte's equation.
Event C * : v is not on the boundary of t; Event R * k : v is at a distance k to the right of e on the boundary of t; (0 ≤ k ≤ p); Event L * k : v is at a distance k to the left of e on the boundary of t. (0 ≤ k ≤ q). These events, as well as the discussion below, are illustrated in Figure 4.
When the event C * occurs, the unexplored part of (t, σ), denoted u, is again a bicolored triangulation of polygon. If * = +, then u has the boundary condition + p+2 -q and the numbers of monochromatic edges and internal faces in u are respectively |E(t, σ)| and |F(t)|− 1. It follows that for all (t 0 , σ 0 ) ∈ BT p+2,q , In other words, P p,q+1 (C + ) = t z p+2,q z p,q+1 and conditionally on C + , the law of u is P p+2,q . Similarly when * = -, we have P p,q+1 (C -) = νt z p,q+2 z p,q+1 and conditionally on C -, the law of u is P p,q+2 . When the event R + k occurs for some 0 ≤ k ≤ p, the vertex v is on the + boundary of (t, σ), and the unexplored part is made of two bicolored triangulations of polygons joint together at the vertex v. We denote by u the right one and by u the left one. Then u has the boundary condition + k+1 and u the boundary condition + p+1−kq . Again one can relate the numbers of monochromatic edges and of internal faces in u ∪ u to E(t, σ) and F(t). It then follows that for all (t , σ ) ∈ BT k+1,0 and (t , σ ) ∈ BT p+1−k,q , In other words, and conditionally on R + k , the maps u and u are independent and follow respectively the laws P k+1,0 and P p+1−k,q .
Similarly, one can work out the probabilities that the events R - In each case, the unexplored part consists of two bicolored triangulations of some polygons which are conditionally independent and follow the law of Boltzmann Ising-triangulations of appropriate Dobrushin boundary conditions (See Figure 4). Tutte's equation simply expresses the fact that the probabilities of the events C + , In each line on the right hand side of this equation, the last term corresponds to the case where t is the edge map, which is a special case that does not belong to any of the events above. The negative term is needed to compensate for the fact that R * p and L * q actually represent the same event. Multiplying both sides by z p,q+1 yields the following recurrence relation, valid for all p, q ≥ 0: where p 1 , p 2 , q 1 , q 2 are summed over non-negative values. Summing the last display over p, q ≥ 0, we get Tutte's equation satisfied by Z(u, v). By exchanging u and v we obtain another functional equation of Z. The two equations can be written compactly as the following linear system.
x denotes the discrete derivative with respect to the variable x ∈ {u, v}. Geometrically, the other equation in the system describes the removal of a boundary edge with spin + next to the origin. This linear system will be the starting point of the asymptotic analysis of the double sequence (z p,q ) p,q≥0 in Section 3. But let us first turn our attention to the geometric implications of the above derivation of Tutte's equation and define the peeling process mentioned in the introduction.

Peeling exploration of the left-most interface
The peeling process along the left-most interface I is constructed by iterating the facerevealing operation used in the derivation of Tutte's equation. Formally, we define the peeling process as an increasing sequence (e n ) n≥0 of explored maps. At each time n, the explored map e n consists of a subset of faces of (t, σ) containing at least the external face and separated from its complementary set by a simple closed path. We view e n as a bicolored triangulation of a polygon with a special uncolored internal face (not necessarily triangular) called the hole. It inherits its root and its boundary condition from (t, σ). The complementary of e n is called the unexplored map at time n and denoted u n . It is a bicolored triangulation of a polygon (without holes). 3 Notice that u n may be the edge map, in which case e n is simply (t, σ) in which an edge is replaced by an uncolored digon. However, this may only happen at the last step of the peeling process (see below). We have seen in Figure 4 that revealing an internal face on the boundary splits (t, σ) into one or two unexplored regions delimited by closed simple paths. To iterate this facerevealing operation, one needs a rule that chooses one of the two unexplored regions, when there are two, as the next unexplored map. At first glance, the natural choice would be to  Figure 5: An example of the n-th and the (n + 1)-th steps of a peeling process. The unexplored map u n is rooted at ρ n , similarly for u n+1 . The peeling step S n+1 is R + 10 rather than L + 6 because we choose to fill the unexplored region on the right.
keep the unexplored region containing ρ † , the end point of the interface I. However, this choice does not fit well with the limit q → ∞, p → ∞ that we would like to take. Instead, we choose the unexplored region with greater number of -boundary edges (in case of a tie, choose the region on the right). This guarantees that when q = ∞ and p < ∞, we will automatically choose the unbounded region as the next unexplored map. We apply this rule inductively to build the peeling process starting from u 0 = (t, σ). At each step, the construction proceeds differently depending on the boundary condition of u n : (i) If u n has a non-monochromatic Dobrushin boundary condition, let ρ n be the boundary vertex of u n with a -on its left and a + on its right (ρ 0 = ρ). Then u n+1 is obtained by revealing the internal face of u n adjacent to the boundary edge on the left of ρ n and, if necessary, choose one of the two unexplored regions according to the previous rule. Figure 5 gives a possible realization of the peeling process in this case.
(ii) If u n has a monochromatic boundary condition of spin -, then we choose the boundary vertex ρ n according to some deterministic function A of the explored map e n , called the peeling algorithm, which we specify later in Section 5. We then construct u n+1 from u n and ρ n in the same way as in the previous case.
(iii) If u n has a monochromatic boundary condition of spin + or has no internal face (i.e. it is the edge map), then we set e n+1 = (t, σ) and terminate the peeling process at time n + 1.
We will explain why the above construction defines the peeling exploration of the left-most interface in Section 6. By induction, u n always has a Dobrushin boundary condition. As mentioned in the introduction, (P n , Q n ) denotes the boundary condition of u n , and (X n , Y n ) = (P n − P 0 , Q n − Q 0 ). Also, S n denotes the peeling event that occurred when constructing u n from u n−1 , which takes values in the set of symbols The above quantities are all deterministic functions of the bicolored triangulation (t, σ). We view them as random variables defined on the sample space Ω = BT = p,q BT p,q .
According to the discussion in the derivation of Tutte's equation, under the probability P p,q and conditionally on (P n , Q n ), the unexplored map u n is a Boltzmann Ising-triangulation of the (P n , Q n )-gon -this is called the spatial Markov property of P p,q . In particular, the couple (P n , Q n ) determines the conditional law of S n+1 in the same way as (p, q) determines the law of S 1 , and the peeling event S n+1 determines the increment (P n+1 − P n , Q n+1 − Q n ) in the same way as S 1 determines (X 1 , Y 1 ). It follows that: (i) Both (P n , Q n ) n≥0 and (X n , Y n ) n≥0 are adapted to the filtration generated by (S n ) n≥1 .
(ii) (P n , Q n ) n≥0 is a Markov chain under P p,q , which we recall is the law of (S n ) n≥1 under P p,q . Its transition probabilities can be deduced from Table 1.
(iii) The functional relation (S n ) n≥1 → (X n , Y n ) n≥0 does not depend on (p, q).
Notice that the law P p,q is completely determined by the data in Table 1, and in particular is independent of the peeling algorithm A. In particular all our results on the limit of P p,q and of the perimeter processes are independent of the peeling algorithm. The choice of A will only become important in the construction of the local limits P p and P ∞ , and will be specified in Section 5.2. This independence reflects the invariance of the law of a Boltzmann Ising-triangulation with monochromatic boundary condition under the change of origin. A similar observation was made for the peeling of non-decorated maps in [17].
In order to study the limits of P p,q , let us first solve Tutte's equation and derive the asymptotics of (z p,q ) p,q≥0 stated in Theorem 1.

Solution of Tutte's equation
Inverting the matrix on the right hand side of (2), we obtain the following system of equations: Remark that both equations are affine in Z. Solving the first one gives the following expression of Z as a rational function of the univariate series Z 0 and Z 1 :

Elimination of the first catalytic variable
It turns out one can obtain a closed functional equation for Z 0 (u) by coefficient extraction. More precisely, by extracting the coefficients of v 0 and v 1 in (3) and (4), seen as formal where we write ∆Z i = ∆ u Z i (u) and z i = z i,0 = z 0,i for short. Notice that only (9) contains the unknown Z 3 , so it can be discarded without loss. On the other hand, the three remaining equations are linear in (Z 1 , Z 2 ). Thus we can easily eliminate these two unknowns to obtain a polynomial equation on Z 0 (u) of the form: (See [1] for details of the elimination) P(Z 0 (u), u, z 1 , z 1,1 ; ν, t) = 0 .
This is not yet a closed functional equation for Z 0 (u) because it involves the series z 1,1 which is a priori not related to Z 0 (u). (It comes from the term ∆ 2 u Z 1 (u) = Z 1 (u)−z 1 −uz 1,1 u 2 in (7).) To relate them, we can view the above equation as a formal power series in u, and extract its coefficients. The first two non-zero coefficients yield two equations relating z 1,1 to z i (i = 1, 2, 3) and which are linear in (z 1,1 , z 2 ). Solving them gives Plugging this intoP = 0 yields a closed functional equation (with one catalytic variable) satisfied by Z 0 (u). This equation can be written as where the rational function R = R(y, u, z 1 , z 3 ; ν, t) is given by (See [1]) Notice that R(Z 0 (u), u, z 1 , z 3 ; ν, t) is a formal power series of t with coefficients in C(ν, u). Therefore (10) determines Z 0 (u) order by order as a formal power series in t. According to the general theory on polynomial equations with one catalytic variable [12,Theorem 3], the generating function Z 0 (u) is algebraic. 4 The same holds for Z 1 (u) and Z(u, v), since according to (6) and (5), they are rational functions of Z 0 (u) and of its coefficients.

Connection with previous work and solution for z i
In principle, we could apply the general strategy developed in [12] to eliminate the catalytic variable u from (10) and obtain an explicit algebraic equation relating z 1 (resp. z 3 ) and t. However, in practice this gives an equation of exceedingly high degree. Instead, we need to exploit specific features of (10) to eliminate u while keeping the degree low. We will explain how this can be done in Appendix A. Here we forego the procedure of eliminating the catalytic variable u and jump directly to the solution of z i (ν, t) (i = 1, 2, 3) by importing the corresponding results from [7]. In [7], the quantity 2Q i (2, ν, t) is the generating series of vertex-bicolored triangulations with a general (i.e. not necessarily simple) boundary of length i and free boundary conditions. The parameter t counts the number of edges and ν the number of monochromatic edges (and the parameter 2 represents the fact that the Ising model is equivalent to the 2-Potts model).
To avoid confusion, we replace the symbols ν and t of [7] by ν * and t * in the following.
Letž i (ν, t) be the generating series of face-bicolored triangulations with a general boundary of length i and monochromatic boundary condition. By using the Kramers-Wannier duality between the low-temperature expansion and high-temperature expansion of the Ising partition function (see e.g. [6, Section 1.2]), one can show that if (ν * , t * ) and (ν, t) satisfy See page 41 in [7] for the details of the computation. On the other hand,ž i (ν, t) is nothing but the version of z i (ν, t) where we remove the constraint of simple boundary. LetŽ 0 (u) ≡ Z 0 (ν, t; u) = 1 + i≥1ži (ν, t)u i . By decomposing a general boundary triangulation into its simple boundary core and general boundary triangulations attached to each boundary vertex of the core, one can show thatŽ 0 (u) = Z 0 (uŽ 0 (u)). This decomposition is known as pruning. It is explained in Figure 6.
Extracting the first coefficients of u, we get Using (12) and (13), we can easily translate the results in [7,Thm. 23] to get the following 4 To apply literally [12, Theorem 3] to (10), we must be able to write R as a polynomial function of the discrete derivatives ∆ i Z 0 (u) (i ≥ 0) and the parameters u, t, which is not obvious here. However, it is clear that the coefficients z 1 , z 2 , z 3 can be written as polynomials of u and ∆ i Z 0 (u) (i ≥ 0). So we can multiply both sides of (10) by u 3 , and view it as a functional equation for the unknown Z 0 (u) = u 3 Z 0 (u). Then u 3 R is clearly a polynomial function of u, t and ∆ i Z 0 (u) = ∆ i+3 Z 0 (u) (i ≥ 0). Therefore [12,Theorem 3] applies. ρ ρ (a) A bicolored triangulation (t, σ) with general boundary.
(b) The simple-boundary core of (t, σ) and the generalboundary triangulations attached to the boundary of the core. Figure 6: By convention, the simple-boundary core of (t, σ) is the component following the root corner of (t, σ) (marked in blue) in the counter-clockwise direction. Pruning consists of decomposing (t, σ) into this simple-boundary core, and one general-boundary component attached to each boundary vertex of the core. As shown in the example, these generalboundary components may be reduced to a single vertex, and this is taken into account by the constant term 1 in the generating seriesŽ 0 (u). For visual clarity, the monochromatic boundary condition is omitted in the drawings.
These rational parametrizations will be checked in the appendix. The singularity analysis of these series can also be imported from [7,Claim 24], which gives Proposition A. One can also give a proof to this theorem using the tools provided in Appendix B.

Singularity analysis at the critical point
To get Z 0 (ν, t; u), the generating function for Ising triangulations with a monochromatic boundary of arbitrary length, we plug the rational parametrization (14) into Equation (10). This gives us an equation of the form E(Z 0 , u; ν, S) = 0 where E is a polynomial of four variables. Under the change of variablesũ = tu andỹ = t u Z 0 (u), we obtain an equation of degree 5 in its main variablesũ andỹ (but of degree 21 overall, see [1]).
It is well known that a complex algebraic curve has a rational parametrization if and only if it has genus zero [31]. Both the genus of the curve and its rational parametrization, when exists, can be computed algorithmically, and these functions are implemented in the algcurves package of Maple. It turns out that the genus of the curve E(Z 0 , u) = 0 is zero, thus a rational parametrization exists. However, the equation is too complicated for Maple to compute a rational parametrization in its full generality in reasonable time. The computation simplifies considerably in the critical case (ν, t) = (ν c , t c ), where t c corresponds to S c = 3 in (14). In this case, we found the following parametrization of Z 0 (u) and the corresponding parametrization of Z 1 (u) deduced from (7): where u = 0 is parametrized by H = 0 and u c = 6 5 (7+ √ 7)t c , as mentioned in Theorem 1. By making the substitution (u, whereẐ(H, K) is a ratio of two symmetric polynomials of degree 10 and 4, respectively. Its expression is given in [1]. Next, we would like to apply the standard transfer theorem of analytic combinatorics [20, Corollary VI.1] to extract asymptotics of the coefficients of Z(u, v). The idea is to use the rational parametrization to write that Z(u, v) =Ẑ(û −1 (u),û −1 (v)) in some neighborhood of the origin, and to extend this relation to the dominant singularity for one of the variables. The main difficulty here is, given a rational parametrization of v → Z(u, v), to localize rigorously its dominant singularity (or singularities), and to show that it has an analytic continuation on a ∆-domain at this singularity. We will present a method that solves this problem in a generic setting in Appendix B. For the sake of continuity of exposition, we first summarize the properties of Z(u, v) and A(u) obtained with this method in the following lemma, and leave its proof to Appendix B. For Notice that a slit disk at x contains a ∆-domain at x.
(iii) For each u ∈ D uc , the function v → Z(u, v) has its dominant singularity at u c and has an analytic continuation on a slit disk at u c (whose margin depends on u).
(iv) Similarly, the function A(u) defined by the rational parametrization in Theorem 1 has its dominant singularity at u c and has an analytic continuation on a slit disk at u c .
Now let us carry out the singularity analysis of Z(u, v) and finish the proof of Theorem 1. By Lemma 5 (ii), the asymptotic expansion of v → Z(u, v) at its dominant singularity u c is determined by the behavior of its parametrization in a neighborhood of K = 1. One can check that the first and second derivatives of K →Ẑ(H, K) both vanish at K = 1. Therefore the function has the Taylor expansion On the other hand, we can rewrite the equation where A(u) is given by the rational parametrization u =û(H) and Thanks to Lemma 5(iii), the transfer theorem [20, This is the last asymptotic stated in Theorem 1. It follows that This can be interpreted as the pointwise convergence of the generating functions of the discrete probability distribution zp,qu p c Zq(uc) p≥0 to the generating function of the sequence apu p c A(uc) p≥0 .
According to a general continuity theorem [20, Theorem IX.1], this implies the convergence of the sequences term by term: for all p ≥ 0. (In fact [20, Theorem IX.1] also assumes the limit sequence to be a probability distribution a priori, but a careful reading of the proof shows that this assumption is not necessary.) Compare the last display with (16), we obtain the asymptotics of (z p,q ) q≥0 stated in Theorem 1. This asymptotics implies in particular that a p ≥ 0 for all p ≥ 0. This positivity property is in fact used in the proof of Lemma 5(iv) in Appendix B. But there is no viscous circle in the proof since we have used only the assertions (i)-(iii) of Lemma 5 to deduce the asymptotics of (z p,q ) q≥0 . Now we repeat the same steps to find the asymptotics of (a p ) p≥0 . Contrary to K →Ẑ(H, K), the first derivative of H →Â(H) does not vanish at H = 1. This leads to an exponent 1/3 instead of 4/3 for the leading order singularity of A(u) at u c : . We apply the transfer theorem again to obtain the asymptotics of (a p ) p≥0 . This completes the proof of Theorem 1.

Limits of the perimeter processes
Let us recall that the peeling process of a bicolored triangulation (t, σ) is an increasing sequence of explored maps (e n ) n≥0 . It is determined by the sequence of peeling events (S n ) n≥1 taking values in the countable set S, plus the initial condition (p, q). We denote by P p,q the law of (S n ) n≥1 when (t, σ) is a Boltzmann Ising-triangulation of the (p, q)-gon.
As stated in Proposition 2, the measure P p,q converges weakly when q → ∞ and then p → ∞. In this section we first prove Proposition 2 and establish the basic properties of the limit distributions P p and P ∞ . Then we move on to prove the scaling limits of the perimeter processes stated in Theorem 3. For convenience, we will denote by L p,q X (resp. L p X and L ∞ X) a random variable which has the same law as the random variable X under P p,q (resp. under P p and P ∞ ).

Construction of P p and P ∞
Since the terms of the sequence (S n ) n≥1 live in a countable space, the weak convergence of P p,q simply means to the convergence of the probabilities of the form P p,q (S 1 = s 1 , · · · , S n = s n ).
In the proof below we will compute explicitly the limits of these probabilities, and verify that the resulting distribution is normalized. Table 2: Law of the first peeling event S 1 under P p , P ∞ and the corresponding (X 1 , Y 1 ).
Lemma 6 (Convergence of the first peeling event). Assume p ≥ 0. The limits exist for all s ∈ S, and we have Proof. The existence of the limits can be easily checked using the expression of P p,q (S 1 = s) in Table 1 and the asymptotics of z p,q in Theorem 1. The explicit expressions of these limits are given in Table 2.
as formal power series in u. With a straightforward (but tedious) calculation using the data in Table 2(a), one can show that the above condition is equivalent to where ∆ u is the discrete derivative operator defined below (2).
. Then one can write down the expansion at v = u c of the second equation in (2), and verify that the coefficient of the dominant singular term (1 − v uc ) 4/3 gives exactly (17). This proves that s∈S P p (S 1 = s) = 1 for all p ≥ 0.
Similarly, using the data in Table 2 This equation can be obtained as the coefficient of (1 − u uc ) 1/3 in the expansion of (17) at u = u c . This completes the proof of the lemma. Proof of Proposition 2. To have the convergence P p,q → P p , we need to define P p (S 1 = s 1 , · · · , S n = s n ) := lim q→∞ P p,q (S 1 = s 1 , · · · , S n = s n ) for all n ≥ 1 and all s 1 , · · · , s n ∈ S. As we have seen at the end of Section 2.2, the peeling events (s k ) 1≤k≤n completely determine the perimeter variations (x k , y k ) 1≤k≤n , independently of the initial condition (p, q). So according to the spatial Markov property, (18) is equivalent to Then Lemma 6 implies that P p is a probability distribution on S Z ≥0 . By Fatou's lemma, this implies that the probability under P p,q of the peeling process stopping in finite time converges to zero. So the peeling process P p -almost surely never stops. Similarly, we take the limit p → ∞ in the above equations, and define P ∞ by The above construction of P p and P ∞ implies immediately the following corollary.
Corollary 7 (Markov property of P p and P ∞ ). Under P p and conditionally on (S k ) 1≤k≤n , the shifted sequence (S n+k ) k≥0 has the law P Pn . In particular, (P n ) n≥0 is a Markov chain. Under P ∞ , the sequence (S n ) n≥0 is i.i.d. In particular, (X n , Y n ) n≥0 is a random walk.

The random walk
The distribution of the first step L ∞ (X 1 , Y 1 ) of this random walk can be readily read from Table 2(b). From there it is not hard to compute explicitly its drift and tails, and deduce Theorem 3(1) by standard invariance principles.
Proof of Theorem 3(1). First, notice that the law of L ∞ (X 1 + Y 1 ) has a particularly simple expression given by It follows that where the derivatives are computed using the chain rule Z 0 (u c ) =Ẑ 0 (1) u (1) and (15). Similarly, we deduce from Table 2(b) and (15) the following expression and value of E ∞ [X 1 ].
We refer to the accompanying Mathematica notebook ([1]) for the computation of the numerical values above. It follows that ]. This is not obvious a priori, since under P ∞ the peeling process always chooses to reveal a triangle adjacent to a + boundary edge, breaking the symmetry between + and -. Again from Table 2 By Theorem 1, their asymptotics is It follows from a standard invariance principle (see e.g. [22,Theorem VIII.3.57]) that the two components of the random walk (X n , Y n ) n≥0 , after renormalization, converge respectively to the Lévy processes X and Y in Theorem 3. Now let us show that these two convergences hold jointly, and that the limits X and Y are independent. We adapt the proof of a similar result for the peeling of a UIPT [15,Proposition 2]. Observe that the steps of the random walk satisfy −2 ≤ max(X 1 , Y 1 ) ≤ 2, so that (X n ) n≥0 and (Y n ) n≥0 never jump simultaneously. Let us decompose (X, Y ) into the sum of two random walks (X (0) , Y (0) ) and (X (1) , Y (1) ) of respective step distributions According to the above observation, (X (0) , Y (0) ) only jumps along the x-axis, and (X (1) , Y (1) ) only jumps along the y-axis. (More precisely, Y k−1 ≤ 2 for all k ≥ 1.) Thus according to the same invariance principle as before, we have in distribution with respect to the Skorokhod topology. Here we have the joint convergence of the two components because the limit of the second component is a constant. Similarly, the random walk (X (1) , Y (1) ) converges to (0, Y t ) t≥0 after renormalization.

Lemma 8 (poissonization and depoissonization)
. Let (W n ) n≥0 be a discrete-time random process in R d (d ≥ 1) and (a n ) n≥0 be a sequence of positive real numbers such that 1 n 1/2 a n sup in probability for all fixed T > 0. If (N t ) t≥0 is a Poisson counting process of intensity 1 and independent of (W n ) n≥0 , then we have In particular, if one of (a −1 n W nt ) n≥0 and (a −1 n W Nnt ) n≥0 converge in distribution with respect to the Skorokhod topology, then the other also converges and has the same limit.
. From the definition of d Dm , we see that the left hand side is bounded by sup where λ is any increasing homeomorphism from [0, m] onto itself. Let λ (n) be the increasing homeomorphism from [0, ∞) onto itself defined by linearly interpolating the function t → n −1 N nt . Then we have W (n) (λ (n) (t)) =W (n) (t) for all t ≥ 0. For each m, we modify λ (n) to produce a homeomorphism λ (n) m from [0, m] onto itself as follows: let t m be the x-coordinate of the point where the graph of the function Using the property of λ (n) and the fact that g m is 1-Lipschitz, we can simplify the bound to get By central limit theorem, √ n sup t≤m |λ (n) m (t) − t| converges in distribution to a finite random variable as n → ∞. Thus the assumption of the lemma implies that the right hand side of the above inequality converges to zero in probability. This completes the proof.

The Markov chain
When p is large, L p (P n ) n≥0 approximates the random walk p + L ∞ (X n ) n≥0 , which has a strictly positive drift µ. This seems to suggest that L p (P n ) n≥0 escapes to +∞ with positive probability (indeed, as P n increases, the transition probabilities of L p (P n ) n≥0 gets closer to those of p + L ∞ (X n ) n≥0 ). However, as stated in Theorem 3(2), L p (P n ) n≥0 hits zero with probability one. There is no contradiction because, despite the weak convergence P p → P ∞ , the expectation E p [X 1 ] does not converge to E ∞ [X 1 ] as p → ∞. Actually, we will compute the limit of E p [X 1 ] in the remark after Proposition 11 and see that it is negative.
What happens is that with high probability, the process L p (P n ) n≥0 stays close to the straight line p n = p + µn up to a time of order Θ(p), and then jump to a neighborhood of zero in one single step. The jump occurs because the peeling events of type R ± p+k , for any fixed k ∈ Z, occur with a probability of order Θ(p −1 ) (See Table 2(a)). To formalize this one-jump phenomenon, let us consider the stopping time where m ≥ 0 is some cut-off which will eventually be sent to ∞. In particular, T 0 is the first time that the boundary of the unexplored map becomes monochromatic.
The following lemma gives an upper bound for the tail distribution of T 0 , which implies in particular that the process L p (P n ) n≥0 hits zero almost surely. It will also be used as an ingredient in the proof of Lemma 10.
ap for all p ≥ 1. By Theorem 1, the right hand side decays like p −1 when p → ∞. Hence there exists δ > 0 such that On the other hand, P n increases at most by 2 at each step, therefore P n ≤ p+2n for all n ≥ 0 almost surely under P p . It follows that for all n ≥ 0, for all n ≥ 0. Use the inequality log(1 − x) ≤ −x for 0 < x < 1 and bound the Riemann sum by its integral, we get Now let us quantify the statement that L p (P n ) n≥0 stays close to the line p n = p + µn. In fact, we will formulate the stronger result that with high probability, both L p (X n ) n≥0 and L p (Y n ) n≥0 stay close to x n = µn up to time T m . Fix some arbitrary > 0. For n ≥ 0, let Then, define the stopping time , then we would have T m = ∞ almost surely, and τ x < ∞ with high probability in the limit x → ∞ thanks to the law of iterated logarithm for heavy-tailed random walks [30]. The following lemma affirms that we can still use the function xf (n) to bound the deviation of L p (X n , Y n ) n≥0 up to time T m in the limit p → ∞ and when both x and m are large.
Lemma 10 (One jump to zero). For all > 0, The proof of Lemma 10 is based on technical estimates on the transition probabilities of the Markov chain L p (P n , Y n ) n≥0 and is left to Appendix C. Now let us complete the proof of Theorem 3 (2). We have seen that L p (P n ) n≥0 hits zero almost surely in finite time. It remains to show that its scaling limit is the process Proposition 11 below ensures that the time T m of the big jump has ζ as scaling limit when p → ∞, regardless of the value of m. Therefore to prove Theorem 3(2) it suffices to show that the process L p (p −1 P pt ) t≥0 converges to 1 + µt before time p −1 T m , and to zero after time p −1 T m .
According to the definition of τ x , for all n < τ x , the distance between p −1 P n and 1 + µn is bounded uniformly by xf (τ x )/p. Lemma 10 and Proposition 11 together ensure that with high probability we have τ x = T m and T m is of order p. This implies that the distance between p −1 P n and 1+µn converges uniformly to zero on n < T m with probability arbitrarily close to 1 when p → ∞ and for x, m large enough. On the other hand, by the spatial Markov property, the shifted process L p (P Tm+n ) n≥0 has the same distribution as (P n ) n≥0 with some initial condition P 0 supported on {0, . . . , m}. Since the distribution of the process depends on p only through the initial condition P 0 and that P 0 is supported on a finite set, the rescaled process L p (p −1 P Tm+ pt ) t≥0 converges identically to zero when p → ∞. This proves Theorem 3(2) provided that Proposition 11 is true.
Proposition 11. For all m ∈ N, the jump time T m has the same scaling limit as follows: Proof. First observe that T 0 ≥ T m , so by strong Markov property, In particular, P p (T 0 − T m > p) − −− → p→∞ 0 for all m ∈ N and > 0. This explains why the scaling limit of p −1 T m does not depend on m.
The rest of the proof is basically a refinement of the estimate of P p (T 0 > tp) given in Lemma 9. The idea is that, before time T m , the Markov chain (P n ) n≥0 stays close to the line P n = p + µn. Therefore at time n there is a probability roughly P p+µn (P 1 ≤ m) to jump below level m at the next step. On the other hand, from Table 2(a) we can read the exact expression of P p (P 1 ≤ m) and show that for all m ≥ 0, there is a constant c m such that Then (20) is obtained by summing the above estimate over all steps up to time tp.
More precisely, let us fix x > 0, m ∈ N and ∈ (0, µ). Take p large enough so that P palmost surely, τ x ≤ T m . Let E = {τ x < T m } be the event of small probability in Lemma 10, where the (P n ) n≥0 deviates significantly from p + µn before jumping close to zero (E for "exceptional"). Also let N n = {τ x > n} be the event that the trajectory of (P n ) n≥0 stays close to the line p n = p + µn up to time n (N for "normal"). Obviously (N n ) n≥0 is a decreasing sequence. Moreover, one can check that On event N n , we have |P n − (P 0 + µn)| ≤ xf (n). Combine this with the asymptotics of P p (P 1 ≤ m), we obtain that for P 0 = p large enough, .
Combine these estimates with the two inclusions in (21), we obtain that on the one hand, And on the other hand, Notice that N n ⊂ {T m > n} ⊂ N n ∪ E up to a P p -negligible set. Thus we have by induction From the Taylor series of the logarithm we see that for all Therefore for any positive sequence (x n ) n≥0 , On the other hand, in the limit p → ∞ we have We also have tp n=0 ( cm+ p+µn−xf (n) ) 2 − −− → p→∞ 0. Combine this with the last three displays, we conclude that Now take the limit m, x → ∞. The last term on the right tends to zero thanks to Lemma 10. The middle terms lim inf P p (T m > tp) and lim sup P p (T m > tp) do not depend on m because of the limit P p (T 0 − T m > p) − −− → p→∞ 0 seen at the beginning of the proof. Moreover, the increasing sequence (c m ) m≥0 has a limit c ∞ . Thus by sending → 0, we obtain Now it remains to show that in fact we have c ∞ = 4 3 µ. Using c m = lim p→∞ p P p (P 1 ≤ m) and the data in Table 2(a), c ∞ can be written as The probabilities can be read from Table 2(a), which gives for all k ≥ 0. Then we can evaluate explicitly c ∞ using Theorem 1 and the asymptotics (16). After a tedious calculation with several miraculous factorizations by the end, we obtain The right hand side can be evaluated using the rational parametrization of A(u), and we find indeed c ∞ = 1 3 √ 7 = 4 3 µ. Remark. (i) We remarked at the beginning of the section that the limit of E p [X 1 ] when p → ∞ should be negative. One can actually compute this limit using the value of c ∞ in the above proof, as follows: first, write E p [X 1 ] as the sum The random variable in the first term is compactly supported, so the convergence in distri- In the second term, the value of X 1 is contained in [−p, −p+m], while we have Using the exact distribution of X 1 in Table 2, it is not hard to bound the third term and show that it converges to zero as p → ∞ and m → ∞. Therefore lim With our approach, it is quite amazing to find such a simple exponent 4/3 for the scaling limit of the jump time T m . Currently we do not have any explanation of this exponent apart from the computation above. Going one step back, one can see that the value 4/3 relies on the algebraic identity More importantly, we expect the same phenomenon to appear in any reasonable model of critical Ising-decorated maps, because the exponent 4/3, which describes the believed scaling limit of an Ising-decorated map, ought to be universal. In a work in progress, we have checked that this is indeed the case when we consider Boltzmann Ising-triangulations with spins on the vertices. It would be very interesting to have an algebraic or probabilistic explanation of this universality.

Local convergence of Boltzmann Ising-triangulations
In this section we construct the local limit of the finite Boltzmann Ising-triangulations when q → ∞ and p → ∞. Both the construction and the proof of the convergence rely on the peeling process. More precisely, a finite Ising-triangulation can be encoded by its peeling process (e n ) n≥0 , which in turn is encoded by its sequence of peeling events (S n ) n≥0 as described in Section 2.2. We have seen in Section 4.1 that the distribution P p,q of the peeling events (S n ) n≥0 converges towards the limits P p and P ∞ .
To recover the local limit of the original Ising-triangulations, we will try to invert the above encoding. Namely, we will try to recover the sequence of explored maps (e n ) n≥0 from the peeling events (S n ) n≥0 , and then to recover the infinite Ising-triangulation (t, σ) from the sequence of finite maps (e n ) n≥0 . The first step is straightforward and will be carried out in the next paragraph under both P p and P ∞ . The second step is significantly more technical and requires different treatments under P p and under P ∞ . This will be the subject of the rest of this section. We summarize the relations between the above objects in Figure 7. Recall that we denote by L p,q X (respectively by L p X and L ∞ X) a random variable having the same distribution as X under P p,q (respectively under P p and P ∞ ). With a slight abuse, we extend this notation to random variables defined under P p,q and under the to-be-constructed measures P p and P ∞ .

Convergence of the peeling process
Definition of L p (e n ) n≥0 and L ∞ (e n ) n≥0 . We will treat the two cases in a unified way by fixing some p ∈ N ∪ {∞}. To recover the sequence of explored maps (e n ) n≥0 from the peeling events (S n ) n≥0 , one only needs to know the initial condition e 0 and the finite Ising-triangulations which are possibly swallowed at each step.
For e 0 , consider Z with its usual nearest-neighbor graph structure and canonical embedding in the complex plane. We view it as an infinite planar map rooted at the corner at the vertex 0 in the lower half plane. The upper-half plane is its unique internal face and is a hole. Then L p e 0 is defined as the deterministic map Z in which a boundary edge has spin + if it lies int he interval [0, p] and spin -otherwise.
Let (u * p,q,n )p ,q,n≥0 be a family of independent random variables which are also independent of (S n ) n≥0 , such that u * p,q,n is a Boltzmann Ising-triangulation of the (p,q)-gon. Under P p,q , one can recover the distribution of e n as a deterministic function of e n−1 , S n and (u * p,q,n )p ,q≥0 . For example, when S n = Rk with some k ≤ P n−1 , then one reveals a triangle in the configuration Rk in the unexplored region of e n−1 , and uses u * k,1,n to fill into region swallowed by this new face. The result has the same law as e n under P p,q . We define L p (e n ) n≥0 by iterating the same deterministic function on L p e 0 , L p (S n ) n≥0 and (u * p,q,n )p ,q,n≥0 . Let F n be the σ-algebra generated by e n . Then the above construction defines a probability measure on F ∞ = σ(∪ n F n ), which we denote by P p by a slight abuse of notation.
Convergence towards L p (e n ) n≥0 and L ∞ (e n ) n≥0 Since (u * p,q,n )p ,q,n≥0 has a fixed distribution and is independent of (S n ) n≥0 , Proposition 2 implies that L p,q (S n ) n≥0 and (u * p,q,n )p ,q,n≥0 converge jointly in distribution when q → ∞ and p → ∞. Here we are considering the convergence in distribution with respect to the discrete topology, namely, for any element ω choose peeling algorithm A glue independent copies of L 0 (t, σ) indicates that the object B is constructed from A. The label "discrete" over the solid arrows indicates that the convergences take place with respect to the discrete topology on the first n terms of the sequences.
in the (countable) state space of the sequences (S n ) n≥0 and (u * p,q,n )p ,q,n≥0 up to time n 0 < ∞, . A caveat here is that the initial condition L p,q e 0 does not converge in the above sense, simply because L p,q e 0 is deterministic and takes a different value for each (p, q). However, for any positive integer K, the restriction of L p,q e 0 (respectively, L p e 0 ) on the interval [−K, K] does stabilize at the value that is equal to the restriction of L p e 0 (respectively, L ∞ e 0 ) on [−K, K]. With this observation in mind, let us consider the truncated map e • n , obtained by removing from e n all boundary edges adjacent to the hole, as in Figure 8. It is easily seen that the number of remaining boundary edges is finite and only depends on (S k ) k≤n . It follows that for each n fixed, e • n is a deterministic function of (S k ) k≤n , (u * p,q,k )p ,q≥0;k≤n and e 0 restricted to some finite interval [−K, K] where K is determined by (S 1 , . . . , S n ). As the arguments of this function converge jointly in distribution with respect to the discrete topology (under which every function is continuous), the continuous mapping theorem implies that for all bicolored map b and for all integer n ≥ 0. The following lemma says that one can replace n in the above convergence by any finite stopping time.
Lemma 12 (Convergence of the peeling process). Let F • n be the σ-algebra generated by e • n . If θ is an (F • n ) n≥0 -stopping time that is finite P p -almost surely, then for all bicolored map b, The same statement holds when P p,q and P p are replaced by P p and P ∞ , respectively.
Proof. First assume that the map b is finite. Since the state of the explored region uniquely determines the past of the peeling process, for every fixed b, there exists some finite n = n(b) Since θ is an (F • n ) n≥0 -stopping time, the event {θ = n} is a measurable function of e • n . Therefore {e • n = b} ∩ {θ = n} is either empty or equal to {e • n = b}. Hence (23) follows from (22). Obviously e • θ is finite if and only if θ is. By Fatou's lemma, summing (23) over the finite maps b gives lim inf q→∞ P p,q (θ < ∞) ≥ P p (θ < ∞) = 1. It follows that lim q→∞ P p,q (θ = ∞) = 0 In particular, (23) also holds when b is infinite (the right hand side is zero).
The same proof goes through when P p,q and P p are replaced by P p and P ∞ respectively.
Remark. Notice that we have not yet specified the peeling algorithm A, which chooses the initial vertex of the peeling in the case of a monochromatic -boundary. This means that the results up to this point are valid for any choice of A.

Convergences towards P p
Although the convergences of peeling processes L p,q e • n → L p e • n and L p e • n → L ∞ e • n are proved exactly in the same way, the local convergence of the underlying random triangulation is much simpler in the first case, namely P p,q → P p . As mentioned after Theorem 4, this is thanks to the fact that, the peeling process (e n ) n≥0 eventually explores the entire triangulation almost surely under P p , provided one chooses an appropriate peeling algorithm. In this section we will specify one such algorithm A, use it to construct P p , and then prove the local convergences P p,q − −− → q→∞ P p and P p,(q 1 ,q 2 ) − −−−− → q 1 ,q 2 →∞ P 0 in Theorem 4. In the introduction we sketched the definition the local distance on the set BT of bicolored triangulations of polygon. Now let us expand it in more details and in the general context of colored maps, so that the definition also applies to objects like the explored maps e n , e • n or the balls in them.

Local limit and infinite colored maps. For a map
for all r ≥ 0 and all balls b of radius r.
When restricted to the bicolored triangulations of the polygon BT , the above definitions construct the corresponding set BT \ BT of infinite maps. Recall from Section 1 that BT ∞ is the set of infinite bicolored triangulation of the half plane, that is, elements of BT \ BT which are one-ended and have an external face of infinite degree.  fills into the hole to give (t, σ). We denote by ∂e n , called the frontier at time n, the path of edges around the hole in e n .
For all r ≥ 0, let θ r = inf {n ≥ 0 : d en (ρ, ∂e n ) ≥ r}, where d en (ρ, ∂e n ) is the minimal graph distance in e n between ρ and vertices on ∂e n . It is clear that this minimum is always attained on the truncated map e • n , therefore d en (ρ, ∂e n ) is F • n = σ(e • n )-measurable and θ r is an (F • n ) n≥0 -stopping time. Expressed in words, θ r is the first time n such that all vertices around the hole of e n are at a distance at least r from ρ. Since (t, σ) is obtained from e n by filling in the hole, it follows that [t, σ] r = [e • θr ] r for all r ≥ 0. In particular, the peeling process (e n ) n≥0 eventually explores the entire triangulation (t, σ) if and only if θ r < ∞ for all r ≥ 0.
Recall that in our context of peeling along the left-most interface, the peeling algorithm is used to choose the origin ρ n of the unexplored map u n when its boundary ∂e n is monochromatic of spin -. (See Section 2.2.) Under P p , we can ensure θ r < ∞ almost surely for all r ≥ 0 with the following choice of the peeling algorithm A: let ρ n = A(e n ) be the left-most vertex on ∂e n that realizes the minimal distance d en (ρ, ∂e n ) from the origin. The idea is that whenever ∂e n is monochromatic of spin -, the peeling process tries to peel off the faces closest to the origin. But by Lemma 9, the number of + edges on ∂e n drops to zero infinitely often P p -almost surely, so that every face will eventually be covered. More precisely: Lemma 13. θ r is finite P p -almost surely for all r ≥ 0 and p ≥ 0.
Proof. The almost surely statements in this proof are with respect to P p . We have θ 0 = 0. Assume that θ r < ∞ almost surely for some r ≥ 0. Then the ball [t] r is also almost surely finite. For t ≥ θ r , let v t be the left-most vertex in [t] r \ [t] r−1 that remains on the frontier ∂e t at time t. Then at every time n ≥ t such that ∂e n becomes monochromatic with spin -, we have A(e n ) = v t . By construction, the next peeling step peels the edge immediately on the left of v t . Since S n+1 has the law of L 0 S 1 , the vertex v t is swallowed at time n + 1 with a fixed non-zero probability conditionally on F n . By Lemma 9, the frontier ∂e n becomes monochromatic of spin -almost surely in finite time, and hence infinitely often by the spatial Markov property. Therefore the above construction implies that every vertex of [t] r \ [t] r−1 is swallowed by the peeling process almost surely in finite time. It follows that θ r+1 < ∞ almost surely.
By induction, θ r is finite almost surely for all r ≥ 0.
Definition of P p . Lemma 13 implies that for every fixed r the sequence ([e n ] r ) n≥0 stabilizes P p -almost surely for n large enough. We define the infinite Boltzmann Ising-triangulation of law P p by its finite balls L p [t, σ] r := lim n→∞ L p [e n ] r . Since every finite subgraph of (t, σ) is covered by e n for n large enough P p -almost surely, its complement only have one infinite connected component, namely the one containing the unexplored map u n . Therefore L p (t, σ) is almost surely one-ended. The external face of L p (t, σ) obviously has infinite degree. So it is indeed an infinite bicolored triangulation of the half plane.
Proof of the convergence P p,q dloc − −− → q→∞ P p . The (F • n )-stopping time θ r is almost surely finite under P p,q and P p , and [t, σ] r = [e • θr ] r is a measurable function of e • θr . Thus it follows from Lemma 12 that P p,q ([t, σ] r = b) − −− → q→∞ P p ([t, σ] r = b) for all r ≥ 0 and all ball b. This implies the local convergence P p,q − −− → q→∞ P p .
Proof of the convergence P p,(q 1 ,q 2 ) dloc − −−−− → q 1 ,q 2 →∞ P 0 . Recall that P p,(q 1 ,q 2 ) is the law of L p,q 1 +q 2 (t, σ) after its origin is translated q 1 edges to the left along the boundary, see Figure 10. Since the peeling process follows the left-most interface, it is not affected by the translation of the origin up to T 0 , the time when the left-most interface is completely explored. It follows that L p,(q 1 ,q 2 ) e T 0 has the same law as L p,q 1 +q 2 e T 0 up to the change of origin. So Lemma 12 implies that after removing the origin, in distribution with respect to the discrete topology. As in Figure 10, let E be the number of edges of ∂e T 0 which are not on the boundary of (t, σ). Also, let S 1 (resp. S 2 ) be the number of -boundary edges swallowed by e T 0 on the right (resp. left) of the origin. It is clear that (E, S 1 , S 2 ) is a measurable function of e • T 0 which does not depend on the position of the origin. Thus the above convergence in law of e • T 0 implies that L p,(q 1 ,q 2 ) (E, S 1 , S 2 ) − −−−− → q 1 ,q 2 →∞ L p (E, S 1 , S 2 ) in law. As shown in Figure 10, the perimeter of u T 0 satisfies Q T 0 = E + (q 1 − S 1 ) + (q 2 − S 2 ). So we have L p,(q 1 ,q 2 ) Q T 0 → ∞ and thus P 0,Q T 0 → P 0 weakly in probability as q 1 , q 2 → ∞. By the spatial Markov property, P 0,Q T 0 is the law of u T 0 conditionally on F T 0 . It follows that in distribution. For a fixed r ≥ 0, the ball [t, σ] r differs from [u T 0 ] r only if the latter contains one of the edges counted by E. These edges are at a distance at least min(q 1 − S 1 , q 2 − S 2 ) from the origin along the boundary of u T 0 . As q 1 , q 2 → ∞, this distance goes to ∞ in probability whereas L p,(q 1 ,q 2 ) [u T 0 ] r converges to L 0 [t, σ] r in distribution. Thus the probability that [u T 0 ] r differs from [t, σ] r converges to zero when q 1 , q 2 → ∞. Then it follows from (24) that for all r ≥ 0 and ball b, Figure 10: Definition of (E, S 1 , S 2 ) in the proof of P p,(q 1 ,q 2 ) dloc − −−−− → q 1 ,q 2 →∞ P 0 .

Definition of P ∞
Recall that θ r is the first time n that the explored map e n covers the ball of radius r in (t, σ), so that [e • n ] r = [t, σ] r for all n ≥ θ r . By definition, it is a stopping time with respect to the filtration F • n = σ(e • n ) defined above Lemma 12. We have seen that, with an appropriate choice of the peeling algorithm, θ r is finite P p -almost surely. This implied that r for all r defines a bicolored triangulation e • ∞ of the half plane.
(ii) If P p is the law of the bicolored triangulation in (i), then P p,q dloc − −− → q→∞ P p in distribution. In Section 4.2 we have seen that the perimeter processes (X n ) n≥0 and (Y n ) n≥0 drift to +∞ almost surely under P ∞ . In particular they are bounded from below, that is, some vertices on the boundary of (t, σ) are never reached by the peeling process. Therefore, the analog of (ii) cannot be true for the limit P p dloc − −− → p→∞ P ∞ . However, we will show that the analog of (i) still holds. The resulting Ising-triangulation L ∞ e • ∞ , called the ribbon for reasons that shall be clear later, corresponds to the region in L ∞ (t, σ) that is eventually explored by the peeling process. It will be glued to other pieces of maps to construct L ∞ (t, σ).

Construction of the ribbon
To prove the analog of (i), one needs to check that the sequence (L ∞ [e • n ] r , n ≥ 0) stabilizes in finite time for all r ≥ 0, and that the resulting infinite bicolored triangulation L ∞ e • ∞ is one-ended, almost surely.
n ] r , n ≥ 0) stabilizes in finite time}. If r 0 < ∞, then there exists a vertex v on the boundary of the ball lim n→∞ L ∞ [e • n ] r 0 such that the peeling process reveals infinitely many edges incident to v. By inspection of the possible peeling steps, one can see that when a new edge incident to v is revealed, the distance between ρ n and v along the frontier ∂e n is at most 2. (Recall that ρ n is the vertex where the + and -parts of ∂e n meet.) This implies that, if the peeling process revealed infinitely many edges incident to v, then either (X n ) n≥0 or (Y n ) n≥0 would visit the same level infinitely many times. We know that this is not the case P ∞ -almost surely. Therefore r 0 = ∞ almost surely, that is, (L ∞ [e • n ] r , n ≥ 0) stabilizes in finite time for all r ≥ 0, and L ∞ e • ∞ is well defined. For each n, consider the graph e • ∞ \ e • n consisting of all the edges and vertices incident to a triangle revealed after time n. Almost surely under P ∞ , the frontier ∂e n has both + and -spins for all n. One can check that in this case the triangles revealed by two consecutive peeling steps always share an vertex, therefore e • ∞ \ e • n is connected. On the other hand, the argument in the previous paragraph shows that for every vertex v, no face incident to v is revealed after some finite time. Hence if V is the complement of some finite subset of vertices of e • ∞ , then V contains the vertices of e • ∞ \ e • n for n large enough. It follows that V can have only one infinite connected component, namely the one containing e • ∞ \ e • n . This proves that L ∞ e • ∞ is almost surely one-ended.
Definition of P ∞ . The reasons for choosing the following notations will be clear in the next subsection. Let R ∞ = e • ∞ be the ribbon under P ∞ , and denote by P 0 the image of P 0 by the inversion of spins. Let L ∞ u ∞ and L ∞ u * ∞ be two random variables of law P 0 and P 0 , respectively, such that L ∞ u ∞ , L ∞ u * ∞ and L ∞ R ∞ are mutually independent. The boundary of L ∞ R ∞ is partitioned into three intervals: one finite interval consisting of edges of e 0 , and the two infinite intervals on its left and on its right. We glue L ∞ u ∞ (resp. Figure 11: The construction of P ∞ . L ∞ u * ∞ ) to the left (resp. right) interval as in Figure 11. Since each piece is one-ended and the gluing between any two pieces occurs at infinitely many edges, the resulting bicolored triangulation is also one-ended. We call P ∞ its law. It is clear that L ∞ (e n ) n≥0 is indeed the peeling process of a random bicolored triangulation of law P ∞ .

Convergence of the ribbon
We have defined the Ising triangulation L ∞ (t, σ) as the disjoint union of the ribbon L ∞ R ∞ and the two unexplored maps L ∞ u ∞ and L ∞ u * ∞ . To prove the local convergence P p → P ∞ , we would like to partition the Ising tringulation L p (t, σ) into three disjoint parts which converge locally to L ∞ R ∞ , L ∞ u ∞ and L ∞ u * ∞ , respectively. After that, we can use the fact that gluing two locally converging maps of the half plane along their boundary result in a locally converging map (see Lemma 15).
However, since the peeling process eventually explores the map L p (t, σ) entirely, there is no canonical way to define the ribbon R ∞ under P p . Instead, let us fix some arbitrary m > 0 and define R m to be the explored map e • Tm−1 plus the triangle revealed at time T m . With this definition, R m , u Tm and u * Tm form a partition of the Ising-triangulation (t, σ) under P p , where u * Tm is the triangulation swallowed by the peeling step at T m . Since we are interested in local limits with respect to the vertex ρ, we will reroot the unexplored map u Tm close to ρ, more precisely at the vertex ρ u as shown in Figure 12. With the notation of Theorem 4, the boundary condition of u Tm is of the form (P, (Q 1 , Q 2 )), with Q 2 = ∞. Similarly, we root u * Tm at the vertex ρ u * as in Figure 12 and denote its boundary condition by ((P * 1 , P * 2 ), Q * ). Recall that T m is the first time n ≥ 0 such that P n ≤ m. By inspection of the possible peeling events, one can confirm that P n may decrease only when S n is of type R + k or Rk . Thus the condition S Tm ∈ {R + P (Tm−1) +Km , R -P (Tm−1) +Km } uniquely defines an integer K m . As shown in Figure 12, K m represents the position relative to ρ † of the vertex where the triangle revealed at time T m touches the boundary.
We want the triple L p (R m , u Tm , u * Tm ) to converge in distribution to L ∞ (R ∞ , u ∞ , u * ∞ ) with respect to the local topology. However this cannot be true without further amendment, because for any fixed m, there is always a non-vanishing probability that the large jump of the process (X n ) n≥0 occurs before T m . (For example, we have τ x = T m+1 < T m , i.e. the large jump arrive at X = m + 1 instead of X = m, with some positive probability.) Instead, Figure 12: The ribbon R m in the case (a) K m ≥ 0, or (b) K m ≤ 0. The unexplored map u Tm on the left of the ribbon (i.e. outside) is rooted at ρ u and has boundary condition (P, (Q 1 , Q 2 )), with Q 2 = ∞. The unexplored map u * Tm on the right of the ribbon (i.e. inside) is rooted at ρ u * and has boundary condition ((P * 1 , P * 2 ), Q * ). We encode the spin of the triangle revealed at time T m by δ ∈ {0, 1}. The sign ≈ means equal up to a difference of 1 depending on δ.
we can only say that the convergence in distribution takes place on some event of large probability. This is formulated as follows.

Lemma 14 (Convergence of the ribbon). For fixed
where E is any set of triples of balls.
Remark. (i) Under P ∞ , the integer K m is not well-defined, while T m = ∞ almost surely. So the event {τ x < ∞} on the right hand side of (25) is essentially J c under P ∞ .
(ii) If J had probability one under both P p and P ∞ , then the right hand side of (25) would vanish, and (25) would express exactly the local convergence in distribution locally in distribution and that (R m ,ũ Tm ,ũ * Tm ) = (R m , u Tm , u * Tm ) on the event J , then (25) will follow. This is roughly how we will show (25) in the proof below.
Proof. As in the statement of Lemma 14, we fix the numbers , x, m > 0 and drop them from the notation of quantities depending on them. Recall that τ x is the first time that either X n or Y n violates the bounds For some N 0 large enough, the left and the right hand side of the above inequality are strictly increasing in n for n ≥ N 0 . Let us define (N r ) r≥0 inductively by for all r ≥ 0. In other words, N r+1 is the first time that the lower bound at time N r+1 exceeds the upper bound at time N r . Assume N r+1 < τ x . Then there exists a time n r ∈ (N r , N r+1 ] such that X nr < min n∈(nr,τ x ) X n , that is, at time n r the process (X n ) n≥0 visits (−∞, X nr ] for the last time before τ x . See Figure 13(a). Geometrically, this means that the triangle revealed at time n r stays on the + boundary of e • n up to time τ x − 1. For the same reason, there is an n r ∈ (N r , N r+1 ] such that the triangle revealed at time n r stays on the -boundary of e • n up to time τ x − 1. As shown in Figure 13(b), the above discussion implies that if N r+1 < τ x , then by the time N r+1 , the peeling process must have covered e • Nr by at least one layer of explored triangles spreading continuously from the + boundary to the -boundary of e • τ x −1 . On the event J , we have τ x = T m , thus R m is by definition equal to e • τ x −1 plus one triangle. It follows that e • N r+1 contains all the vertices at a distance 1 from e • Nr with respect to the graph distance inside R m . By induction, we have Since N r is solely determined by x and , the previous condition is always satisfied on the event J ⊆ {τ x ≥ p}, for any fixed r and for p large enough.
Next let us find a simple bound for the boundary conditions of u Tm and u * Tm on the event J . The boundary condition ((P * 1 , P * 2 ), Q * ) of u * Tm can be related to the perimeter processes by considering the following quantities (see Figure 12): P * 1 + P * 2 + Q * = P Tm−1 + K m + 1 , (total perimeter of u * Tm ) S + + P * 2 − min(0, K m ) = p , (number of edges between ρ and ρ † ) (number of -edges on the boundary of u * Tm ) After rearranging the terms, we get P * 1 = X Tm−1 + S + + δ , P * 2 = p − S + + min(0, K m ) and Q * = max(0, K m ) + (1 − δ) . On the event J = {τ x = T m ≥ p} ∩ {K m ≤ m} and for p large enough, we have and where δ is either 0 or 1, depending on the peeling step that reveals the vertex ρ u * (see Figure 12). It follows that and Q * ≤ m + 1 on J . Notice that P * 1 → ∞ and P * 2 → ∞ when p → ∞. Similarly, one can show that R m n r n = n r Figure 13: (a) Before time τ x , the perimeter processes (X n ) n≥0 and (Y n ) n≥0 stay between the increasing barriers µn ± xf (n). Thus one can define a deterministic sequence of times N r up to τ x , such that X Nr is smaller than the minimum of X n on [X N r+1 , τ x ). It follows that between N r and N r+1 , both X n and Y n must visit some level for the last time before τ x . (b) Since the peeling process explores consecutively the triangles along the Ising interface, between the last visit times n r and n r , it must have explored a continuous layer of triangles spreading between the left and the right boundaries of e • τ x −1 . On the event {τ x = T m }, one can replace e • τ x −1 by R m .
on J for some deterministic number Q 1 = Q 1 ( , x, m, p) such that Q 1 − −− → p→∞ ∞. Consider two random bicolored triangulationsũ Tm andũ * Tm such that conditionally on R m , they are independent Ising-triangulations of respective boundary conditions (P∧(m+1), (Q 1 ∨Q 1 , ∞)) and ((P * 1 ∨P * 1 , P * 2 ∨P * 2 ), Q * ∧(m+1)). Thanks to the estimates in the previous paragraph, we have on the event J . (More precisely, there is a suitable coupling between the two sides such that the equality holds.) According to (22), we have L p e • Nr − −− → p→∞ L ∞ e • Nr with respect to the discrete topology. On the other hand, since Q 1 ∨ Q 1 → ∞ uniformly when p → ∞, and P ∧ (m + 1) takes values in the finite set {0, . . . , m + 1}, the convergence P p,(q 1 ,q 2 ) (Remark that the proof of P p,(q 1 ,q 2 ) → P 0 in Section 5.2 also works when q 2 = ∞.) Similarly, L pũ * Tm − −− → p→∞ L ∞ u * ∞ locally in distribution. These two convergences takes place conditionally on R m , and the limits do not depend on R m . It follows that we have the joint convergence where three components on the right hand side are mutually independent, as prescribed by the definition of P ∞ . Equations (26) and (27) imply respectively and lim On the event {τ x = ∞} we have [e • Nr ] r ⊆ R ∞ almost surely with respect to P ∞ , so Then (25) follows from (28), (29) and (30) by the triangle inequality.

Convergence towards P ∞
The triangulation L p (t, σ) (respectively, L ∞ (t, σ)) can be seen as the result of gluing the triple L p (R m , u Tm , u * Tm ) (respectively, L ∞ (R ∞ , u ∞ , u * ∞ )) along their boundaries. To deduce P p → P ∞ from Lemma 14, one wants to show that local convergence is preserved by this gluing operation. First, let us look into the simpler setting of gluing two maps at their roots.
To keep familiar notations, let us consider probability measures P p (p ≥ 0) and P ∞ on some probability space Ω. Let m and m be two colored, possibly infinite random maps defined on Ω. Assume that m and m always have simple boundaries, and that L ∞ m and L ∞ m are almost surely maps of the half plane. Denote by ρ and ρ the root vertices of m and m . Let L be a random variable on Ω taking positive integer or infinite values, such that Finally, let m ⊕ m be the map obtained by gluing the L boundary edges of m on the right of ρ to the L boundary edges of m on the left of ρ . The dependence on L is omitted from this notation because the local limit of m ⊕ m is not affected by the precise value of L, provided that (31) is true. The following lemma affirms this claim, and relates the local convergence of m ⊕ m to the local convergence of m and m .
Lemma 15 (Gluing of locally convergent maps). Let ε ≥ 0. If for all r ≥ 0 and all sets E, then m ⊕ m satisfies the same inequality, that is, for all r ≥ 0 and all sets E, Remark. When ε = 0, the lemma says that if the couple L p (m, m ) converges jointly in distribution to L ∞ (m, m ) with respect to the local topology, then so does their gluing m⊕m .
For > 0, one should interpret (33) as saying that L p (m ⊕ m ) converges locally in distribution to L ∞ (m ⊕ m ) on some event of probability at least 1 − . Similarly for (32).
Proof. For all r ≥ 0, let G r be the σ-algebra generated by ([m] r , [m ] r ). The assumption (32)  . Let R r be the minimal radius R ≥ 0 such that the above condition is satisfied. Then the event {R r ≤ R} is in G R and for any set of balls E, the intersection {[m ⊕ m ] r ∈ E} ∩ {R r ≤ R} is also G R -measurable. That is, R r is a G-stopping time and the event {[m ⊕ m ] r ∈ E} is in G Rr . Also, R r < ∞ almost surely because the union of the balls [m] R and [m ] R eventually covers the whole map m ⊕ m when R → ∞.
For any R ≥ 0, let Since E R and E R are G R -measurable, the assumption of the lemma yields lim sup The right hand side tends to when R → ∞ since R r < ∞ almost surely. This gives (33).
In the above reasoning we have ignored the possibility that L, the total number of glued edges in m ⊕ m , may be smaller than R. Taking this into account adds an extra error term of lim sup p→∞ P p (L ≤ R) + P ∞ (L ≤ R) to the right hand side of the last display. But this term is zero if L satisfies the assumption (31). This completes the proof of the lemma.
The Ising-triangulation (t, σ) is obtained either by gluing u Tm and u * Tm to R m under P p , or by gluing u ∞ and u * ∞ to R ∞ under P ∞ . To be precise, one needs to move the root vertex of R m before each gluing: Given a map m with a simple boundary, and an integer S, let us denote by − → m S (resp. ← − m S ) the map obtained by translating the root vertex of m by a distance S to the right (resp. to the left) along the boundary. The proof of the following lemma is left to the reader. Lemma 16 (Local convergence is preserved by a finite translation of the root). Assume that S is almost surely finite under P p and P ∞ . If L p m → L ∞ m locally in distribution jointly with L p S → L ∞ S as p → ∞, then L p − → m S also converges to L ∞ − → m S locally in distribution.
Under P p , we have and S + and Sare the distances from ρ to ρ u * and ρ u , respectively. See Figure 12. Similarly, L ∞ (t, σ) can be expressed in terms of u ∞ , R ∞ , u * ∞ and S ± using gluing and root translation. On the event J , the perimeter processes (X n ) n≥0 and (Y n ) n≥0 stay above the barrier µn−xf (n) up to time τ x . Thus their minima over [0, τ x ) are reached before the deterministic time N min = sup {n ≥ 0 : µn − xf (n) ≤ 0} and S + and Sare measurable functions of the explored map e • N min . It follows that L p S ± converges in distribution to L ∞ S ± on the event J , in a sense similar to (32) and (33) in Lemma 15. This convergence also takes place jointly with the one in Lemma 15. Using the relation (34), it is not hard to adapt the proof of Lemma 15 and deduce from Lemma 14 the local convergence of the Ising-triangulation (t, σ) on the event J , in the following sense.

Corollary 17.
Fix any x, m, > 0. Then for all radius r ≥ 0 and set E of balls, we have The left hand side does not depend on the parameters x, m and used to define the ribbon R m and the event J . Therefore to conclude that P p converges locally to P ∞ , it suffices to prove that: For the probability of J c , we use the union bound The first two terms on the right can be bounded using Lemma 10 and Proposition 11: For the last term, let us first fix some n ≥ 1 and consider P p (τ x = T m = n and K m > m). Since g x, := max n≥0 xf (n) − µn is finite, for large p we have p + µn − xf (n) ≥ p − g x, > m for all n ≥ 0. It follows that τ x ≤ T m and {τ x = T m = n} = {τ x > n − 1 and P n ≤ m}. Notice that the event {τ x > n − 1} is F • n -measurable and P n−1 ≥ p − g x, on that event. Hence by the spatial Markov property, Summing over n ≥ 1 and then taking the limit p → ∞ gives lim sup The number P 1 is determined by S 1 . More precisely, from the relation between S 1 and X 1 in Table 2 one can see that Thus the right hand side of the above inequality can be rewritten as provided that the limits in the numerator and the denominator exist and commute with the summations. The existence of the limits can be verified directly using the data in Table 2: One can also compute their sum over k ≥ 0 and check that it commutes with the limit: It follows that (35)  This completes the proof.
Remark. The upper bound of P p (τ x = T m and K m > m) in the proof of Lemma 18 can be refined to the following identity in the limit p → ∞: for any fixed m ≥ 0 and k ≥ 0, the random variables K m and δ (the latter is defined in Figure 12) satisfy The limits on the left hand side of the above equalities define a probability distribution supported on {(k, δ) ∈ Z × {0, 1} : δ − k ≤ m}, where the condition δ − k ≤ m comes from the fact that P Tm = δ + max(0, −K m ) ≤ m. One can further take the limit m → ∞ in the above equalities. The result defines a probability distribution given by normalizing the weights w(k, δ) on (k, δ) ∈ Z × {0, 1}, where w(k, 1) = lim p→∞ p · P p (S 1 = R + p+k ) and w(k, 0) = lim p→∞ p · P p (S 1 = R -p+k ), or explicitly, We interpret this distribution as the distribution of the peeling event immediately after the large jump of the perimeter process (P n ) n≥0 in the infinite Ising-triangulation of law P ∞ . (Of course, the large jump P ∞ -almost surely never occurs. So this is only an interpretation.) Recall that our peeling process explores the triangles adjacent to the left-most interface from ρ to ρ † . This set of triangles is invariant in distribution when ρ and ρ † swap their roles. For this reason, the distribution in the last paragraph should be related to the distribution of S + and S -(see Figure 12) under P ∞ . The derivation of the exact relation, though conceptually straightforward, is very tedious and will not be carried out here.

Properties of the interfaces and spin clusters
In this section, we discuss some properties of the interfaces and spin clusters in the infinite Ising-triangulations of the laws P p and P ∞ which are direct consequences of our construction of P p and P ∞ . These include the statements (2) and (3) of Theorem 4. Firs,t let us take a closer look at the definition of the spin clusters and their relation to the interfaces.

Vertex-connected clusters and edge-connected clusters
Since in our model the spins are on the faces of the triangulation, there are two equally natural definitions of the spin clusters. Two faces can be considered adjacent as soon as they share a vertex, or they can be considered adjacent only when they share an edge. The resulting connected components of faces of the same spin will be called vertex-connected clusters in the first case, and edge-connected clusters in the second case. Obviously vertex-connected clusters are larger than their edge-connected counterparts. Notice that an edge-connected cluster of spin -is surrounded by vertex-connected clusters of spin +, and vise-versa, see Figure 15(a).
Notice that we have not specified the type of the infinite clusters in Theorem 4 (2)(3). By this we mean that the two statements are valid for both the edge-connected clusters and vertex-connected clusters. The same applies to the following discussion.
Cluster structure of bicolored triangulations with a Dobrushin boundary condition By convention, we shall consider consecutive boundary edges of the same spin to be in the same cluster, as in Figure 15(a). This implies that, in a bicolored triangulation with a non-monochromatic Dobrushin boundary condition, there will be exactly one cluster containing the + boundary edges, and one cluster containing the -boundary edges. All the other clusters are non-adjacent to the external face.
As shown in Figure 15(b), the left-most interface I from ρ to ρ † separates the edgeconnected cluster containing the -boundary from the vertex-connected cluster containing the + boundary. Similarly, the right-most interface from ρ to ρ † separates the vertex-connected cluster of containing the -boundary from the edge-connected cluster containing the + boundary. On the other hand, a spin cluster that does touch the boundary has an outer-most interface, as highlighted in the example in Figure 15(b).
Peeling process along the left-most interface Recall that, when the boundary of the unexplored map is not monochromatic, we defined the peeling process to reveal the triangle adjacent to the -boundary edge on the left of the root ρ n of the unexplored map. As shown in Figure 16(b), as long as the revealed triangle has spin -and does not swallow ρ n , the peeling process turns around the vertex ρ n and does not extend the Ising interface. When the peeling process reveals a triangle of spin + incident to ρ n , the Ising interface is extended by one edge which is the left-most non-monochromatic edge adjacent to ρ n . Therefore our peeling process indeed explores the triangulation along the left-most Ising interface from ρ.
When the boundary of the unexplored map becomes monochromatic -(that is, when n = T 0 ), the peeling process chooses some triangle on its boundary to reveal (according to the peeling algorithm A) until the boundary becomes non-monochromatic again. In terms of the clusters, this means that after exploring the entire left-most interface from ρ to ρ † , the peeling process wanders into the bulk of the edge-connected cluster containing theboundary, and waits until the first time that it encounters again a triangle ∆ of spin +. After that, the peeling process turns around the vertex-connected cluster containing the triangle ∆ in the clockwise direction. It finishes exploring it when the boundary of the unexplored map becomes monochromatic -again.
The above observations on the relation between peeling process and the cluster structure allow us to deduce Theorem 4 (2)(3) from what we know about the perimeter processes.
Proof of Theorem 4 (2)(3). Under P p , the stopping time T 0 is almost surely finite, therefore The left-most interface I (white) and the right-most interface (yellow) from ρ to ρ † in a bicolored triangulation with Dobrushin boundary condition. A vertex-connected cluster not touching the boundary is also shown. Its outer-most interface is highlighted in red.
(t, σ) ∼ P ∞ I Figure 16: (a) Position of the ribbon (shadowed region) relative to the spin clusters in the Ising-triangulation of law P ∞ . The peeling process follows the left-most interface I from ρ. the left-most interface I from ρ to ρ † is finite. Since the Ising-triangulation of law P p is one-ended, it follows that the vertex-connected cluster containing the + boundary is finite almost surely. Similarly by the Markov property, the perimeter process (P n ) n≥0 hits zero infinitely often almost surely, which shows that every vertex-connected cluster of spin +, as shown in Figure 15(b), is almost surely finite. This proves Theorem 4 (2). Under P ∞ , the Ising-triangulation is composed of two copies of L 0 (t, σ), the Isingtriangulation of law P 0 (with a spin inversion in one of them), and a ribbon consisting of triangles adjacent to the left-most interface I, see Figure 16(a). As shown in the previous paragraph, there is exactly one infinite cluster in each copy of L 0 (t, σ). In the ribbon, there are two infinite clusters (one of each spin) along the two sides of its boundary. However, since the ribbon is glued to the copies of L 0 (t, σ) along infinitely many edges, the two infinite clusters in the ribbon almost surely merges with the infinite clusters in the copies of L 0 (t, σ) after gluing, thus leaving only two infinite clusters in the Ising-triangulation of law P ∞ . The fact that the ribbon touches the boundary only in a finite interval is due to the positive drift of the perimeter processes (X n ) n≥0 and (Y n ) n≥0 .
Peeling process along the right-most interface When constructing the peeling process, we could have chosen to reveal the triangle adjacent to the + boundary edge on the right of ρ n instead of the -boundary edge on the left of ρ n . By symmetry, this would define a peeling process along the right-most interface from ρ (see Figure 16(c)). Under P p , this new peeling process would explore the boundary of the edge-connected cluster containing the + boundary edges, as shown in Figure 15(b). Under P ∞ , the new peeling process would explore the boundary separating the infinite vertex-connected cluster of spin -from the infinite edge-connected cluster of spin + (Figure 16(d)).
This change from left to right will change the law of the first peeling event S 1 and the relation between S 1 and (X 1 , Y 1 ) in Table 1, thus changing the law of the peeling process and the perimeter processes. However, the results in Theorem 1, 3, 4 and Proposition 2 will not change except for the value of the constants b, c x and c y . Their proofs can also be carried out in the same way. We leave the reader to check the above claim by constructing the counterpart of Table 1 and carrying out calculations using the data in it.
Interestingly, peeling along the right-most interface gives a different construction of the law P ∞ , which splits the infinite triangulation with a different ribbon. The relation between the ribbon in the old construction and the ribbon in the new construction is illustrated in Figure 16(a,d). Of course, the two constructions yield the same result because they both construct the local limit of P p,q when q → ∞ and then p → ∞.
Under a global spin inversion and a mirror reflection of the triangulation, an Isingtriangulation of law P p,q becomes an Ising-triangulation of law P q,p and the left-most interface in the former is mapped to the right-most interface in the latter. Therefore our claim that the peeling along the left-most interface and the peeling along the right-most interface defines the same law P ∞ implies that P ∞ is also the local limit of P p,q when p → ∞ and then q → ∞. This is one of the facts that support the conjecture that P p,q → P ∞ when p, q → ∞ at any relative speed.
Perimeter of the clusters More quantitative properties of the clusters in the infinite Ising-triangulations can also be derived from the construction of P p and P ∞ . In the rest of this section we will discuss the relation between the perimeter processes (X n , Y n ) n≥0 and the actual perimeter of the spin clusters.
We have seen that when the boundary of the unexplored map is non-monochromatic, the peeling process explores the perimeter of spin clusters: either the left-most interface of the cluster containing the + boundary, or the outer-most interface around a cluster of spin + not touching the boundary. More precisely, each peeling step contributes additively to the length of the perimeter being explored, and conditionally on the sequence (S n ) n≥1 of peeling events, the contribution of the different steps are independent random variables.
Let η p,q denote the total length of the left-most interface I from ρ to ρ † in an Isingtriangulation of law P p,q . The contribution to the length η p,q made by each peeling event is summarized in Figure 17. Notice that when S n = L + k or S n = Rk , the peeling step swallows a region that contains the interface being explored, and the law of the contribution to the total length is given by η 1,k or η k,1 . It follows that η p,q satisfies the following equation in distribution: where the random variables on right hand side are taken under P p,q , and F (S n ) is 0, 1, or an independent random variable with the law of η 1,k or η k,1 , determined according to Figure 17. The last peeling step along I occurs at time T 0 . Its contribution to the total length depends on S T 0 in a different way than the previous steps. We leave the interested reader to work out its exact distribution G(S T 0 ).
The above discussion is also valid when q = ∞. If we assume in addition that p is large, then the contribution of the last step S T 0 to the total length of the left-most interface will be negligible, and η p ≡ η p,∞ satisfies When p → ∞, Proposition 11 states that p −1 L p T 0 has a limit in distribution. Moreover, the terms in the above sum converge in distribution to an i.i.d. sequence of law F (L ∞ S 1 ). Recall that P ∞ (S 1 = L + k ) ∼ c 1 k −7/3 and P ∞ (S 1 = Rk ) ∼ c 2 k −7/3 when k → ∞, for some constants c 1 and c 2 (see Table 2(b)). Therefore according to Figure 17, the random variable F (L ∞ S 1 ) has a finite expectation if and only if k≥0 (k + 1) −7/3 E[η 1,k + η k,1 ] < ∞. If this is indeed the case, then the perimeter η p will have a scaling limit similar to the one of L p T 0 : Figure 17: The contribution to the length of the interface by the 6 types of peeling events.

Proposition 19.
Assume that k≥1 (k + 1) −7/3 E[η 1,k + η k,1 ] < ∞, then the total length η p of the left-most interface in L p (t, σ) has the scaling limit We will not prove this claim in this paper. Its proof is an adaptation of the proof of Proposition 11. While we do not have a proof of the first moment condition in Proposition 19, we can verify a similar condition for the Ising-triangulation with a general (i.e. not necessarily simple) boundary. Since the finiteness of the expected perimeter should be a geometric property of the scaling limit of the model, by universality, we believe that the same property also holds for the current model.

A Elimination of the second catalytic variable in Tutte's equation
In Section 3.1 we showed how to eliminate one of the two catalytic variables (u, v) in Tutte's equation by extracting appropriate coefficients of the series Z(u, v). In the end we obtained an algebraic equation with one catalytic variable of the form where R(y, u, z 1 , z 3 ; ν, t), given explicitly by (11), is a polynomial in y u , u, z 1 , z 3 , t and ν. To eliminate the second catalytic variable u, we use a generalization of the quadratic method used by Tutte in his study of properly colored triangulations [34,35]. It is later adapted in [7, Section 12] to treat bicolored maps with monochromatic boundary condition. In our setting, this method consists of finding two rational functions J(u, y), L(u, y) and a polynomial C(x) whose coefficients do not depend on u or Z 0 (u), such that (10') can be written in the form A · L(u, Z 0 (u)) 2 = C(J(u, Z 0 (u))) where A is some polynomial that may depend on all the variables. Then the square factor on the left hand side would suggest that C(x) has a double root, in the same way as the classical quadratic method (see e.g. [21, Section 2.9]). With some trial-and-errors, we discovered the following choice of J and L: L(u, y) = 2 ty u + (ν + 1)J(u, y) .
Notice that the mapping (u, y) → (J, L) is invertible. Thus we can make the reverse change of variable and rewrite (10') as a polynomial equation satisfied by the variables J and L, with coefficients in the space of formal power series C(ν) [[t]]. As shown in [1], we obtain the following equation as the result: where L = L(u, Z 0 (u)), J = J(u, Z 0 (u)), and C 2 , C 0 are the following polynomials with coefficients in C(ν)[[t]]: Notice that z 3 → w is just a linear change of variable for fixed z 1 . Now we derive heuristically an algebraic equation satisfied by z 1 and t. We will check a posteriori that they lead to the right solution. We can write (36) in two ways: If we view t and J as two independent variables, and view L as a function of (t, J). Then the above equations suggest that both C 0 and C 0 + C 2 2 , viewed as polynomials of J, have double roots. It is well known that this is characterized by their discriminants being zero. The discriminant D 2 = 0 provides an equation that relates z 3 (t) to z 1 (t) and t. Under the change of variablesz 3 = t 9 z 3 ,z 1 = t 3 z 1 andt = t 2 and after removing irrelevant factors, it gives a quadratic equation forz 3 . In [1] we check that this equation, as well as the equation of degree 6 relatingz 1 tot, are both satisfied by the rational parametrizations (14).

B Singularity analysis via rational parametrization
In this section we present a method to locate the dominant singularity of a combinatorial generating function from a proper rational parametrization of it. First let us clarify the definition of a rational parametrization.

Definition.
Let E ∈ C[x, y] be an irreducible polynomial. A couple of rational functions P = (x,ŷ) is an (affine) rational parametrization of the curve E(x, y) = 0 if E(x(s),ŷ(s)) = 0 for all but finitely many s ∈ C. Here a rational function is seen as a continuous mapping from C to C. The rational parametrization P is • real ifx andŷ can be written with real coefficients.
• proper if P(s) = (x, y) has a unique solution s for all but finitely many (x, y) on E = 0.
(iii) If s c is the only critical point of P such that |x(s)| = x c , then there exists a neighborhood V of s 0 such that s c ∈ ∂V andx| V is a conformal bijection from V onto a slit disk at x c . Moreover, φ has an analytic continuation on this slit disk.
Proof. (i) The existence and uniqueness of s 0 is guaranteed by Lemma 20. But since P and φ are real,s 0 is also a solution to the problem. So we have s 0 =s 0 by uniqueness.
(ii) Up to the change of variable s ← −s, we can assume thatx (s 0 ) > 0. Let s c = inf{s ≥ s 0 :x (s) = 0 orŷ(s) = ∞} and x c =x(s c ), thenŷ • (x −1 ) is an analytic continuation of φ on [0, x c ). By Pringsheim's theorem, the radius of convergence of φ is at least x c . It is well known that the only entire functions that satisfy algebraic equations are polynomials. Therefore x c < ∞ =x(∞) according to the hypothesis that φ is not polynomial. It follows that s c < ∞ and s c is a critical point of P.
If φ is analytic at x c , then by analytic continuation, the relationŷ = φ •x holds in a neighborhood of s c , i.e. (P, s c ) parametrizes φ locally at x c . This contradicts Lemma 20 (ii). Therefore x c is a dominant singularity of φ(x). Recall that we derived in Section 3.3 a rational parametrization of Z of the form (u, v) = (û(H),û(K)) and Z =Ẑ(H, K). We obtain a rational parametrization of u → Z(u, u) by taking K = H: where Q is some polynomial of degree 6. In And it follows that p,q≥0 z p,q u p v q is absolutely convergent for all (u, v) ∈ D 2 uc . On the other hand, if the series is absolutely convergent for some (u, v) with |u| > u c , then by monotonicity the series Z 0 (u) = Z(u, 0) will have a radius of convergence strictly larger than u c . This is not the case because the rational parametrization (15) implies that Z 0 (u) has a singularity of type (u c − u) 4/3 at u = u c . Now let us fix a u ∈ D uc and prove (iii). Since the coefficients of the series v → Z(u, v) are not necessarily non-negative, Proposition 21 does not apply. Instead, we will check (iii) directly using the formula Z(u, v) =Ẑ(û −1 (u),û −1 (v)) and the analytic properties of the functionû. Recall thatû induces a conformal bijection from some neighborhood V of H = 0 onto a slit disk D | uc at u c , which extends bi-continuously to H = 1 byû(1) = u c . Let U be the preimage of D uc byû| V ∪{1} , then it suffices to show that (iii') for each H ∈ U , K →Ẑ(H, K) has no pole in U \ {1}.
Indeed, since the poles of a univariate rational function are isolated, (iii') implies that K = 1 is the only possible pole of K →Ẑ(H, K) in some neighborhood U of the compact U . Its imageû(U ) is a neighborhood of the disk D uc . Sinceû is a conformal bijection onto D | uc , the composed function v →Ẑ(û −1 (u),û −1 (v)) is analytic on the intersectionû(U ) ∩ D | uc , which contains a slit disk at u c . On the other hand, v → Z(u, v) must have a singularity at u c , otherwise its radius of convergence would be strictly larger than u c , contradicting (i). We conclude that u c is the unique dominant singularity of v → Z(u, v) for all u ∈ D uc .
In order to prove (iii'), we will show the following stronger statement: the denominator ofẐ(H, K) has no zero in C A one-jump lemma for the process L p (X n , Y n ) n≥0 We have seen in the discussion above Lemma 10 that the lemma would become a standard law of iterated logarithm if the process L p (X n , Y n ) n≥0 were replaced by L ∞ (X n , Y n ) n≥0 .
Our proof of Lemma 10 is based on the idea of comparing the transition probabilities of the Markov chain L p (P n , Y n ) n≥0 (recall that P n = p + X n ) to the step distribution of the random walk L ∞ (X n , Y n ) n≥0 , and the fact that P n p → ∞ for all n < T m . The mean technical difficulty is that the convergence L p (X 1 , Y 1 ) → L ∞ (X 1 , Y 1 ) of transition probabilities only implies the convergence of the process (X n , Y n ) n≥0 up to finite time. But we want to estimate probabilities about the behavior of L p (X n , Y n ) n≥0 up to time T m , which is of order Θ(p).
The proof follows the general strategy used in [9] to establish asymptotic behaviors of heavy-tailed random walks. It comes in three steps.
First, we establish two estimates on the step distribution L p (X 1 , Y 1 ): one for the probability that (X 1 , Y 1 ) is far from the origin (Lemma 23(v)) and the other for the exponential moments of (X 1 , Y 1 ), restricted on the event that it remain close to the origin (Lemma 23(viii)). Next, we bound the probability that L p (X n , Y n ) n≥0 deviates to a distance x ≈ χf (N ) from its mean on a time scale N (Lemma 24). The process (X n , Y n ) n≥0 may realize a such deviation either by making a jump of size x, or by accumulating steps of size smaller than x. We use the two estimates in Lemma 23 to bound the probabilities of these two situations. Finally, we complete the proof of Lemma 10 by applying Lemma 24 to an exponentially increasing sequence of time intervals.
To simplify notation, let us write The comparison of the distributions L p (X 1 , Y 1 ) and L ∞ (X 1 , Y 1 ) is based on the following observation: for all k ≤ p − 2, This can be seen by checking in Table 2 that P p (S 1 = s) = a p+X 1 (s) u p+X 1 (s) c apu p c P ∞ (S 1 = s) for every peeling event s ∈ S such that −X 1 (s) ≤ p − 2. (Recall that X 1 and Y 1 are determined by the peeling event S 1 .) If (40) were valid for all k, it would mean that L p (X n , Y n ) n≥0 is a Doob h-transform of the random walk L p (X n , Y n ) n≥0 . However, (40) breaks down for k > p − 2. More precisely, the supports of (X 1 , Y 1 ) under P p and P ∞ differ: as illustrated in Figure 18, the support of L ∞ (X 1 , Y 1 ) is contained in the L-shape defined by −1 ≤ X 1 ∨ Y 1 ≤ 1 (except for one point), whereas the support of L p (X 1 , Y 1 ) stops at X 1 = −p (for the simple reason that P 1 = p + X 1 ≥ 0) and continues in the negative y-direction. We control the probabilities in the part {−X 1 > p − 2} of the support by the crude bound that for e ∈ {0, 1}, P p (−(X 1 , Y 1 ) = (p − e, k )) ≤ P p (−X 1 = p − e) ∼ cst e · p −1 as p → ∞, where the equivalence can be read from Table 2, and was seen in the proof of Proposition 11. The probabilities in the rest of the support is controlled using (40) in conjunction with the following asymptotics, seen respectively in Section 4.2 and in Theorem 1.
The proof of the following properties of and is left to the reader. A 1 B 1 and A   We fix some θ ∈ [ 1 2 , 1) and let p θ = 2 1−θ so that θp ≤ p − 2 for all p ≥ p θ (for example, θ = 1 2 and p θ = 4). Lemma 23. (i) p x k p y k k −7/3 for k ≥ 1.
(viii) First consider the + sign. We decompose the expectation into three terms: The first term is bounded by 1. The second term will be taken care of by (vii). For the last term, notice that W ≤ µ + x on the event A x . We cut the interval (−∞, µ + x) at λ −1 and x/2 and bound the expectation separately on each subinterval: We used the fact that e λW − 1 − λW λ 2 W 2 for W ≤ λ −1 in the first line, and the assumption λ ≤ 1 so that e λµ 1 in the last line. The second inequality in each line follows from (vi). Combining these three bounds with (41) and (vii) gives E p [e λW 1 Ax ] − 1 λx −1/3 + λ 4/3 + e λx/2 λ 4/3 + e λx x −4/3 .
As stated at the beginning of this appendix, we start by considering, instead of τ x , the first time τ x that (X n , Y n ) deviates from its mean for some constant distance x, namely τ x = inf {n ≥ 0 : |X n − µn| ∨ |Y n − µn| > x}. Proof. For n ≥ 1, let ∆X n = X n − X n−1 and ∆Y n = Y n − Y n−1 . Consider J x = inf {n ≥ 1 : (−∆X n ) ∨ (−∆Y n ) ≥ x} , the first time that either (X n ) n≥0 or (Y n ) n≥0 makes a large negative jump of size x. We bound the probability of the event {τ x ≤ N, τ x < T m } separately in the case {J x ≤ τ x } (large jump estimate) and in the case {τ x < J x } (small jump estimate).
Large jump estimate: union bound. Write P p (τ x ≤ N, τ x < T m and J x ≤ τ x ) ≤ P p (J x ≤ τ x ∧ N and J x < T m ) = N n=1 P p (n ≤ τ x and J x = n < T m ) .
On the one hand, J x = n < T m implies that P n > m and (−∆X n ) ∨ (−∆Y n ) ≥ x. On the other hand, we have P n−1 ≥ p − x on the event {n ≤ τ x }. Therefore by the Markov property of L p (P n , Y n ) n≥0 , we have P p (n ≤ τ x and J x = n < T m ) ≤ E p P P n−1 P 1 > m and (−X 1 ) If p ≥p θ = p θ /(1 − θ), x ∈ [1, θp] and p > p − x, then p ≥ p θ . Thus we can use the uniform bound of Lemma 23(v) to bound the above supremum. It follows that for all λ ≥ 0. For p ∈ N ∪ {∞}, let ϕ x,e p (λ) = E p [e λ(µ−X 1 ,µ−Y 1 )·e 1 Ax ], where A x = {(−X 1 ) ∨ (−Y 1 ) ≤ x} is the same event as defined in Lemma 23. Since the couple (X 1 , Y 1 ) takes only finitely many values on the event A x and L p (X 1 , Y 1 ) → L ∞ (X 1 , Y 1 ) in distribution, we have ϕ x,e p (λ) → Let (∆X * n , ∆X * n ) n≥1 be a sequence of i.i.d. random variables independent of (X n , Y n ) n≥0 and such that L p (∆X * 1 , ∆Y * 1 ) = L p * (X 1 , Y 1 ) in distribution. Define By definition, on the event {τ e x = n}, the future (U i , V i ) i>n of the process is an i.i.d. sequence independent of the past such that E p [e λ(µ+U i ,µ+V i )·e 1 {U i ∨V i ≤x} ] = ϕ x,e p * (λ). Therefore we can continue the bound (44) with It is easy to see that τ e x is a stopping time with respect to the natural filtration (F n ) n≥0 of the process (U n , V n ) n≥0 . Therefore for all i ≥ 1, E p e λ(µ+U i ,µ+V i )·e 1 {U i ∨V i ≤x} F i = 1 {i≤τ e x } · ϕ x,e P i (λ) + 1 {i>τ e x } · ϕ x,e p * (λ) ≤ ϕ x,e p * (λ) , where we have the last inequality thanks to the fact that P i ≥ p − x on the event {i ≤ τ e x }. By expanding the expectation in (45) with N successive conditioning, we see that it is bounded by ϕ x,e p * (λ) N . Then we obtain by collecting (44) and (45): P p (τ e x ≤ N, τ e x < J x ) ≤ e −λx (ϕ x,e p * (λ) N ∨ 1) . By Lemma 23(viii), there exists a constant C such that ϕ x,e p (λ) ≤ exp(Cx −4/3 e λx ) for all p ≥ p θ , x ∈ [1, θp], λ ∈ [2x −1 , 1] and unit vector e ∈ Z 2 . As we have seen in the derivation of the large jump estimate, the same bound holds for ϕ x,e p * (λ) = sup p ≥p−x ϕ x,e p (λ), provided that p ≥p θ . Therefore we have P p (τ e x ≤ N, τ e x < J x ) ≤ exp(−λx + C · N x −4/3 e λx ) . Plugging this into (43) and take λx = c log log x with c = 1 + /2 to get .
x k+1 x k x k−1 n ∆ ∆ n = xf (n) N 1 = 2 N 2 = 6 x 1 x 2 Λp N k0 · · · · · · · · · . . . In other words, x k ≤ xf (N k−1 ). Consider the sequence of horizontal segments I k = {(n, x k ) : n ∈ (N k−1 , N k ]} depicted in Figure 19. Thanks to the previous inequality, all of these segments are below the curve ∆ n = xf (n). Let K x,m be the index k where ∆ n goes above I k for the first time up to T m , that is K x,m = inf {k ≥ 1 : ∃n ∈ (N k−1 , N k ] s.t. ∆ n > x k and n < T m } .
Then we have {τ x < T m } ⊆ {K x,m < ∞}. Remark that ∆ N k−1 ≤ x k−1 and ∆ n+N k−1 > x k imply that∆ n := |X n+N k−1 − X N k−1 − µn| ∨ |Y n+N k−1 − Y N k−1 − µn| > ∆x k for any n ≥ 0. Therefore by Markov property of L p (X n , Y n ) n≥0 , P p (K x,m = k) ≤ E p P P N k−1 ∃n ∈ (0, ∆N k ] s.t. ∆ n > ∆x k and n < T m 1 {∆ N k−1 ≤x k−1 } ≤ sup Let k 0 be the largest k such that N k ≤ Λp, where Λ ≥ 1 is some cut-off value that will be sent to infinity after p, x and m. For any fixed x, m and in the limit p → ∞, we have ∆x k−1 ≤ θp and p − x k−1 ≥ p − xf (Λp) >p θ for all k ≤ k 0 . Therefore we can apply Lemma 24 to bound the above supremum, and obtain that P p (K x,m ≤ k 0 ) On the other hand, k 0 < K x,m < ∞ implies that T m > N k 0 . Therefore by Lemma 9, We conclude that for every fixed Λ > 0, and uniformly for x > 0 and m ≥ 1, lim sup p→∞ P p (τ x < T m ) ≤ lim sup p→∞ P p (K x,m < ∞) (log x) − /2 + Λm −1/3 + Λ −γ 0 .
Taking the limit m, x → ∞ and then Λ → ∞ finishes the proof.