Loop percolation on discrete half-plane

We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of percolation at intensity 1/2 was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.


Introduction
We will consider discrete (rooted) loops on Z 2 , that is to say finite paths to the nearest neighbours on Z 2 that return to the origin and visit at least two vertices. The rooted random walk loop measure µ Z 2 gives to each rooted loop of lengths 2n the mass (2n) −1 4 −2n . It was introduced in [8]. In [6] are considered loops parametrized by continuous time rather than discrete time. µ Z 2 has a continuous analogue, the measure µ C on the Brownian loops on C. Let P t z,z ′ (·) be the standard Brownian bridge probability measure from z to z ′ of length t. Let p t (z, z ′ ) be the heat kernel µ C is a measure on continuous time-parametrized loops on C defined as where dz∧dz 2i is the standard volume form on C. The measure µ C was introduced in [9].
Given c > 0 we will denote by L Z 2 c respectively L C c the Poisson ensemble of intensity cµ Z 2 respectively cµ C , called random walk respectively Brownian loop soup. In [8] it was shown that one can approximate L C c by a rescaled version of L Z 2 c . If A is a subset of Z 2 we will denote by L A c the subset of L Z 2 c made of loops contained in A. If U is an open subset of C we will denote by L U c the subset of L C c made of loops contained in U . For δ > 0 we will denote by L A,≥δ c respectively L U,≥δ c the subset of random walk loops L A c respectively Brownian loops L U c made of loops of diameter greater or equal to δ. Similarly we will use the notation L A,≤δ c for the loops of diameter smaller or equal to δ.
We will consider clusters of loops. Two loops γ and γ ′ in a Poisson ensemble of discrete or continuous loops belong to the same cluster if there is a chain of loops γ 0 , γ 1 , . . . , γ n in this Poisson ensemble such that γ 0 = γ, γ n = γ ′ and γ i and γ i−1 visiting a common point. For all c > 0, loops in L Z 2 c as well as in L C c form a single cluster. Thus we will consider loops on discrete half-plane H = Z × N * and on continuous half-plane H = {z ∈ C|Im(z) > 0}, mainly from the angle of existence of an unbounded cluster.
The percolation problem for Brownian loops was studied in [13]. It was shown that for c ∈ (0, 1] L H c has only bounded clusters and for c > 1 the loops in L H c form one single cluster. The problem of percolation by random walk loops was studied in [7], [3] and [12] in more general setting than dimension 2. We will focus on the percolation by loops in L H c . The probability of existence of an infinite cluster of loops follows a 0 − 1 law and there can be at most one infinite cluster ( [12]). Moreover for c = 1 2 loops in L H 1 2 do not percolate ( [12]). This result was obtained through a coupling with the massless Gaussian free field. By considering just the loops that go back and forth between two neighbouring vertices we get a lower bound of clusters of loops by clusters of an i.i.d. Bernoulli percolation. In particular this implies that for c large enough loops in L H c percolate. Hence as the parameter c increases there is a phase transition and a critical value c H * ∈ [ 1 2 , +∞) of the parameter. We will also consider the percolation by the subsets of loops L H,≥δ c and the corresponding critical parameters c H,≥δ * . Using the results on the clusters of Brownian loops from [13] and the approximation result from [8] we will show in section 2 the following: . In [1] there is a proof that in the setting of independent Bernoulli percolation enhancements lower strictly the critical parameter. In our case the diameter of loops is not bounded. Moreover it was observed in [2] that the proof in [1] contains a gap, and it was closed only for limited situations. Yet we conjecture the strict inequality: In section 3 we consider the lattice Z 2 where we add a fixed number of intermediate vertices on each edge and consider the corresponding loop soup as defined in [6]. Let N ≥ 1. Z 2 N be the set The elements of Z 2 N are vertices and the edges join the pairs where j and k are integers. We will also abusively denote Z 2 N the induced graph. See below an illustration of Z 2 4 . Let p (N ) (z, z ′ ) be the transition probabilities of the nearest neighbour random walk on Z 2 N . Let ε ∈ {−1, 1}. The transition probability from ( j N , k) to ( j+ε N , k) or from (k, j N ) to (k, j+ε N ) is 1 2 if j, k ∈ Z and j ∈ N Z, and 1 4 if j ∈ N Z and k ∈ Z. Given z 1 , z 2 . . . , z n ∈ Z 2 N , the measure µ Z 2 N on the loops on Z 2 N gives the mass 1 n p (N ) (z n , z 1 ) to the loop (z 1 , z 2 . . . , z n , z 1 ). Let L   1]. Relying on the fact that the clusters of one-dimensional loops on N undergo a phase transition an c = 1 ([7]), we will prove the following: In section 4 we will again deal with loops on H. We will also consider the loops on whole Z 2 up to a diameter δ, that is to say L Z 2 ,≤δ c . These loops form infinitely many clusters and undergo a phase transition at some parameter c Z 2 ,≤δ * . We will prove the following: Using the theory of enhancements we will deduce from theorem 3 the following: In all three sections 2, 3 and 4 we will consider 1-dependent edge percolations on Z 2 or H, (ω(e)) e edge . By 1-dependent percolation we mean that if two disjoint subsets of edges E 1 and E 2 are at graph distance at least 1 then (ω(e)) e∈E1 and (ω(e)) e∈E2 are independent. According the results on locally dependent percolation by Liggett, Schonmann and Stacey in [10], for all 1-dependent edge percolations on Z 2 or H with p the probability of an edge to be open, there is an universal p(p) ∈ [0, 1) such that the 1-dependent edge percolation contains an i.i.d. Bernoulli percolation with probabilityp(p) of an edge to be open. Moreover the following constrain holds: 2. The critical parameter smaller or equal to 1 Let Q ext and Q int be the following rectangles:     ). One should imagine that the polygonal loops are actually Brownian.
Only a set of loops that is sufficient for the event is represented. Full line loops saty inside Qint Dashed loops cross the boundary of Qint.
We will call the event 3 i=1 C i (L Qext,≥δ c ) special crossing event with exterior rectangle Q ext and interior rectangle Q int . We will also consider translations and rotations of Q ext and Q int and deal with special crossing events corresponding to the new rectangles. We are interested in the event  Next we show that if c > 1 and δ small enough then the probability of is close to 1. (γ(t)) 0≤t≤tγ −→ λ − 1 2 (γ(λt)) 0≤t≤λ −1 tγ So does the Brownian motion B. Thus λT has the same law as T . It follows that T = 0 a.s.
The set of loops L H c \ L Q c is at positive distance from 0 thus B cannot intersect it immediately. It follows that B intersects immediately L Q c . intersects R is the same as a loop in L (−a √ n,a √ n) 2 c of duration comprised between 1 2 and 1 intersects R. This is at least as big as the similar probability for L (0) c . Since the latter probability is non-zero, the intersection events occurs a.s. for infinitely many of L Proof. It is enough to show that the probability of each of the C i (L Qext,≥δ c ) converges to 1 as δ tends to 0. Since the three cases are very similar, we will do the proof only for . Thus there is a chain of loops (γ 0 , . . . , γ n ) in L Qint c , with γ 0 = γ ′ and γ n =γ ′ , joining γ ′ andγ ′ . If δ is the minimum of diameters of (γ 0 , . . . , γ n ) and γ andγ then Let δ > 0. We consider the percolation by Brownian loops on H when the "small" loops are removed, that is to say the clusters of L H,≥δ c . Contrary to L H c , L H,≥δ c has infinitely many clusters for all c > 0, but there is a phase transition for the existence of an unbounded cluster. Moreover the critical value of c does not depend on δ because a change of the value of δ is achieved through a change of scale. We prove that this phase transition happens again at c = 1. We will use a block percolation construction that will also appear subsequently in the proof of theorem 1. Proof. The first statement is trivial since already L H c has only bounded clusters for c ∈ (0, 1]. Assume that c > 1. We consider a depend edge percolation (ω δ (e)) e edge of H on the discrete half plane H. If e is an edge of form c achieves a special crossing event with exterior rectangle (3j − 1, 3j + 2) × (3k − 3, 3k + 3) and interior rectangle (3j, 3j + 1) × (3k − 2, 3k + 2). ω δ is a 1-dependent edge percolation: if two disjoint subsets of edges E 1 and E 2 are such that no edge is adjacent to both E 1 and E 2 , then (ω δ (e)) e∈E1 and (ω δ (e)) e∈E2 are independent. This is due to the fact that the subsets of loops involved in the definition of special crossing events for edges in E 1 and and edges in  Let According to lemma 2.4, if δ is small enough, p δ is close enough to 1 forp(p δ ) to be strictly greater than 1 2 (see the last paragraph of the section 1). Then ω δ contains a supercritical i.i.d. Bernoulli edge percolation on H, and hence an unbounded cluster. It follows that if δ is small enough, L H,≥δ c has a.s. an unbounded cluster. But the probability of an unbounded cluster does not depend on δ. Consequently this is true for all δ > 0.
Next we recall the result on approximation of Brownian loops by random walk loops from [8]. Let N ≥ 1. Consider the rescaled grid 1 N Z 2 and the rescaled Poisson ensembles of discrete loops denote t γ the number of its steps and given a continuous loopγ let tγ its life-time. If γ = (z 0 , z 1 , . . . , z tγ −1 , z 0 ) is a discrete nearest neighbour loop on 1 N Z 2 , define a continuous time parametrized loop Φ N γ as follows: Let θ ∈ ( 2 3 , 2) and r ≥ 1. There is a coupling between L 1 N Z 2 c and L C c such that except on an event of probability at most cste · (c + 1)r 2 N 2−3θ there is a one to one between the two sets such that given a discrete loop γ and the continuous loopγ corresponding to it: made of loops contained in Q ext of diameter greater or equal to δ. This est is finite a.s. The only approximation result that we will need in this section, that we state as a lemma and which follows from what precedes, is Lemma 2.6. Let c > 0 and δ > 0. As N tends to +∞ the random set of interpolating continuous loops We need to show that the above convergence for the uniform norm also implies a convergence of the intersection relations, that is to say that If T r (γ) < +∞ let e iαr := γ(T r (γ)) r Let I j be the real interval For 0 < r 1 < r 2 let A(r 1 , r 2 ) be the annulus A(r 1 , r 2 ) := {z ∈ C|r 1 < |z| < r 2 } For r > 0 let HD(r) be the half-disc We will say that the path γ satisfies the condition C j if 12 2 −j (γ) < +∞, γ hits e i(α 2 −j−1 + π 2 ) I j at a timet j before hitting the circle S(0, 2 −j ) • On the time interval (T 2 −j−1 (γ),t j ) γ stays in the half-disc e iα 2 −j−1 HD(2 −j ) • From timet j the path γ stays in the annulus A( 7 12 2 −j , 9 12 2 −j ) until surrounding once clockwise the disc B(0, 7 12 2 −j ) once clockwise and hitting e i(α 2 −j−1 +π) I j . Figure 5 illustrates a path satisfying the condition C j . If this condition is satisfies than γ disconnects the disc B(0, 7 12 2 −j ) from infinity. Moreover if one perturbs γ by any continuous function f : [0, t γ ] → C such that f ∞ ≤ 1 12 2 −j than the path (γ(s) + f (s)) 0≤s≤tγ disconnects the disc B(0, 2 −j−1 ) from infinity. Moreover the disconnection is made inside the annulus A(2 −j−1 , 2 −j ). s ) 0≤s≤t2 be two independent standard Brownian bridges from z 1 to z 1 and z 2 to z 2 respectively. On the event that b (1) intersects b (2) there is a.s. ε > 0 such that for all continuous functions Proof. Let T (1) 2 be the first time b (1) hits the range of b (2) . If the two path do not intersect each other T satisfies the condition C j for infinitely many values of  is a r.v. defined on the event where b (1) and b (2) intersect. If f 1 and f 2 are such that f i ≤ 1 12 2 − then the path b (1) + f 1 disconnects the disc B(b Let p 0 ∈ (0, 1) such that for all p ∈ (p 0 , 1),p(p) > 1 2 . Let δ be small enough such that p δ > p 0 . Then for all N sufficiently largẽ Then we can define a 1-dependent edge percolation on H, similar to the one used in the proof of proposition 2.5 , using the blocs of loops in L 1 N Z 2 ∩H,≥δ c (see figure  4). Each open edge of this 1-dependent percolation corresponds to the realisation of the three crossings represented in figure 3 by the loops of L 1 N Z 2 ∩H,≥δ c inside a rectangle isometric to Q ext This 1-dependent percolation will contain a supercritical i.i.d. Bernoulli percolation on H, hence the blocs of discrete loops will percolate.

Example of lattices on which the critical parameter is < 1
We consider the loops on the lattice H N (see figure 1). We want to show that for N large enough c HN * The joint distribution of (L c (0), L c (1)) is given by the Laplace transform As N tends to infinity and the intermediate vertices between z and z ′ get more dense, the traces of loops in L HN ,(z,z ′ ) c on the interval joining z to z ′ approximate the one-dimensional Brownian loops on the interval of length 1 joining z to z ′ . See [11], section 3.7. The clusters of L HN ,(z,z ′ ) c restricted to the interval of length 1 joining z to z ′ converge to the clusters of these one-dimensional Brownian loops as well. The ensemble of Brwonian loops has an occupation field (local times) on the interval joining z to z ′ , which is a continuous process, and the clusters of the Brownian loops are delimited by the zero set of the occupation field (see [11], section 4.2, particularly proposition 4.7) The joint distribution of this occupation field at the two endpoints in z and z ′ is given by the Laplace transform (3.1). Coditionally on the two values at extremities, the occupation field is a bridge of a squared Bessel process of dimension 2c (see [11], section 4.2, particularly proposition 4.5). We sum up all this in the following lemma:

Absence of percolation at criticality
We need to prove the following: connects 0 to ∂B(r). To prove proposition 4.1 it is enough to show that there is a continuous function ψ : (0, +∞) 2 → (0, +∞) such that for all r ∈ N * , all c, s > 0 Let L Z 2 ,≤n respectively L Z 2 ,=n+1 be the space of all possible configurations of loops in Z 2 of diameter less or equal to n respectively equal to n + 1. A loop γ in L Z 2 ,≤n or in L Z 2 ,=n+1 will be called r-pivotal for L Z 2 ,n c,s if • L Z 2 ,n c,s does not connect 0 to ∂B(r) • γ does not appear in the Poisson point process L Z 2 ,n c,s • adding γ to L Z 2 ,n c,s creates a connection from 0 to ∂B(r) Let P Z 2 ,≤n (r, c, s) be the random subset of L Z 2 ,≤n made of r-pivotal loops for L Z 2 ,n c,s and let P Z 2 ,=n+1 (r, c, s) be the similarly defined subset of L Z 2 ,=n+1 . Then we have a Russo-type identity: There is an injective map f n from L Z 2 ,≤n to L Z 2 ,=n+1 such that for every γ ∈ L Z 2 ,≤n , the range of γ is contained in the range of f n (γ), and such that f n commutes with the translations, that is to say that if (z 0 , . . . ,zt,z 0 ) = f n ((z 0 , . . . , z t , z 0 )) and v ∈ Z 2 then Proof. Let γ = (z 0 , z 1 , . . . , z tγ , z 0 ) be a loop in L Z 2 ,≤n . We will explain how f n (γ) is constructed. Let e 1 be the unit vector corresponding to the first coordinate. For and by triangle inequality Thus the set {j ∈ N|d j = n + 1} is non-empty. Let j be its minimum. j ≥ 1. We define f n (γ) as the loop (z 0 , z 0 + e 1 , . . . , z 0 + je 1 , z 0 + (j − 1)e 1 , . . . , z 0 , z 1 , . . . , z tγ , z 0 ) That is to say we add a horizontal excursion from z 0 to z 0 + je 1 and back to z 0 . Clearly f n (γ) defined this way belongs to L Z 2 ,=n+1 , its range contains γ and the map commutes with translations. Moreover f n is injective. Indeed to reconstruct γ we only have to erase the first excursion outside the starting point of f n (γ).
Proof of proposition 4.1. We will show the inequality (4.1) using the identities (4.2) and (4.3) and the lemma 4.2. If γ ∈ L Z 2 ,≤n is r-pivotal for L Z 2 ,n c,s and f n (γ) does not appear in L Z 2 ,n c,s then f n (γ) is r-pivotal for L Z 2 ,n c,s too. Thus and a 2 := min Since f n commutes with translations a 2 is well defined and lies in (0, +∞). We have Since f n is injective this means that ∂ ∂s θ r (c, s) ≥ a 2 e −sa1 ∂ ∂c θ r (c, s) Our strategy to prove the theorem 3 will be to show that whenever the loops in L H c percolate, with high probability inside a large rectangular box there is a long crossing connecting two opposite short sides and it is created by random walk loops with an upper bound on their diameter of the same order as the size of the box.
Given c, δ > 0 and A, A 1 , A 2 ⊆ Z 2 we will denote by P A,≤δ c (A 1 ↔ ∂A 2 ) the probability that a cluster of loops in L A,≤δ c connects A 1 and A 2 . We will replace A 2 by ∞ to denote a connection to infinity and erase the exponent ≤ δ if there is no upper bound on diameter of loops. The symbol will signify the absence of connection. We will use the notation A 1 A3 ↔ A 2 to signify the connection between A 1 and A 2 by a path contained in If n ≥ n 0 then According to the approximation from [8] lim It follows that (4.4) is true. The event Λ n0 ↔ ∂Λ n is occurs if and only if one of the events Λ n0 ↔ ∂ l Λ n , Λ n0 ↔ ∂ r Λ n and Λ n0 ↔ ∂ u Λ n does. Moreover all the events are increasing. According to the Harris-FKG inequality By symmetry It follows that which completes the proof.
Next we will deduce that if P H,≤n is close to 1 then with high probability the loops in L Z 2 ,≤n c make a long crossing connecting two opposite short sides of a rectangle of size of order n. We will distinguish between the cases when P H,≤n c (Λ n0 ↔ ∂ l Λ n ) is high and when P H,≤n In the first case we will adapt a standard argument for i.i.d. Bernoulli percolation. In the second case the argument will be specific to our model.
By symmetry and Harris-FKG inequality So it is enough to show that lim inf n→+∞,n≥n0 Let n 1 ≥ n 0 and z 1 , The latter probability is 0 since a.s. L H c has at most one infinite cluster. Thus Proof. Using Harris-FKG inequality we get It suffices to show that (4.5) lim inf n→+∞,n≥n0 If the loops in L Z 2 ,≤n c create inside the box Λ n a path connecting Λ n0 to ∂ l Λ n and a path connecting Λ n0 to ∂ r Λ n , then for the existence of a crossing from ∂ l Λ n to ∂ r Λ n inside Λ n it suffices to have a loop in L Z 2 ,≤n c that surrounds the box Λ n0 . Indeed such a loop will connect the two paths. Moreover, because of the bound on diameter, such a loop will be contained in Λ n . Approximation result of [8] implies that the probability of the existence of such a loop converges as n tends to +∞ to the probability that a Brownian loop in L C,≤1 c surrounds the point 0. According the lemma 4.5 this limit equals 1. Thus (4.5) is true.
Observe, by comparing the sizes of the boxes, that  Proof. The three crossing of the event 3 i=1 C i,n (L Z 2 ,≤4n c ) can be obtained using intersections of crossing inside boxes obtained from Λ 3n by translations and rotations of angle π 2 . This is schematically represented in the next picture where straight lines symbolize crossings. Applying Harris-FKG inequality we get that and the left-hand side above converges to 1 (lemma 4.7).
Proof of theorem 3. We will first prove that if loops in L H c percolate then so do the loops in L Z 2 ,≤4n c for n large enough. Let ω n c be the edge percolation on Z 2 , measurable with respect L Z 2 ,≤4n c , defined as follows: • If e is a horizontal edge {(j, k), (j + 1, k)} then ω n c (e) = 1 if an event similar to Thus for n large enoughp P(ω n c (e) = 1) > 1 2 and ω n c percolates. Conversely let n be an integer and assume that the loops in L Z 2 ,≤n c percolate. According to proposition 4.1 the percolative system L Z 2 ,≤n+1 c is supercritical. For us it is enough to show that loops in L H,≤n+1 c percolate. As the percolative system L Z 2 ,≤n+1 c is only locally dependent (n-dependent), this can be done, for instance, by adapting Grimmett and Marstrand dynamic renormalisation technique ( [5] and [4], section 7.2). We won't detail this.