On the chemical distance in critical percolation

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.


Introduction
The object of this paper is to prove a result concerning the chemical distance inside large open clusters in critical independent bond percolation on Z 2 . The chemical distance between two sets A and B is the minimum number of edges in any lattice path of open edges joining A to B. from the work of Kozma and Nachmias (see also [11,Theorem 2.8] for a more general result, which applies also to long-range percolation). These estimates presumably hold for any dimension above the critical dimension d = 6, but the current proofs rely on results derived from the lace expansion. To the best of our knowledge, there is currently no rigorous work addressing the chemical distance in percolation for 2 < d < 19.
Unlike in the high dimensional case, there has been very little work on the chemical distance in two-dimensional percolation. This is mainly because it is not clear how to analyze it using the usual tools: gluing constructions and arm exponents. Two notable exceptions include the work of J. van den Berg and A. Jarai [31], and G.J. Morrow and Y. Zhang [21], both of which study asymptotics for the lowest crossing. The techniques in our paper are most similar to the ones in these papers, and also particularly the recent work involving arm events in self-destructive percolation by D. Kiss, I. Manolescu and V. Sidoravicius [18]. For extensive background on relevant methods from critical and near-critical percolation, the reader should also see Nolin's survey [23].
As observed by Pisztora [24], the work of M. Aizenman and A. Burchard [1] implies that distances in planar critical percolation are bounded below by a power greater than one of the Euclidean distance, with high probability. Letting B(n) = [−n, n] 2 , there is an > 0 such that, for any κ > 0, P(∃ an open crossing of B(n) with cardinality ≤ n 1+ | ∃ an open crossing) ≤ C κ n −κ . (1.1) For definiteness, we consider horizontal crossings of B(n). Pisztora treats the "nearcritical" case, when the percolation parameter p is sufficiently close to p c = 1 2 and obtains essentially the same result as long as n is below the correlation length for p. H. Kesten and Y. Zhang [16] had previously outlined a proof of an estimate analogous to (1.1) for some fixed κ, for the size of the lowest open crossing in B(n).
We know of no explicit estimate for in (1.1). In principle, such an estimate could be obtained from careful examination of the proof in [1], but the resulting value would be exceedingly small, and it is not likely to correspond to the true typical length of crossings.
In this work we will be concerned with upper, rather than lower bounds for the chemical distance. Conditioned on the existence of a crossing, the obvious approach is to identify a distinguished crossing of B(n) whose size can be estimated. C 1,k n 2k (π 3 (n)) k ≤ EL k n ≤ C 2,k n 2k (π 3 (n)) k , (1.2) with π 3 (n) denoting the "three-arm" probability (see (2.1)). On the triangular lattice, the existence and asymptotic value of the three-arm exponent are known [29], and (1.2) becomes EL k n = n 4k/3+o (1) .
of minimal length. Is it the case thatS n /L n → 0, (1.3) in probability, conditionally on H n ? From [16, p. 603]: "It is not clear thatS n /L n → 0 in probability." In this paper, we give a positive answer to this question. (See Corollary 2.5.) We present our result on the chemical distance in terms of circuits in annuli. The same proof, with minor modifications, applies to the case of horizontal crossings. Let A(n) = B(3n) \ B(n). By Russo-Seymour-Welsh (RSW) [27,30] estimates, the probability that there is an open circuit around B(n) in A(n) is bounded below by a positive number independent of n. Conditioned on the existence of such a circuit, one defines the innermost open circuit γ n as the circuit with minimal interior surrounding B(n) inside A(n). As in the case of the lowest path, one can show that if L n is the size of γ n , then for some C > 0 (1/C)n 2 π 3 (n) ≤ EL n ≤ Cn 2 π 3 (n).
Let S n be the number of edges on the shortest open circuit around B(n) in A(n) (defined to be zero when there is no circuit). Our main result, Theorem 2.2, is the following: as n → ∞, ES n n 2 π 3 (n) → 0. (1.4) This shows that in an averaged sense, S n is much shorter than the typical size of L n .
The formulation (1.4) in terms of circuits in annuli serves as an illustration of the fractal nature of percolation clusters. If macroscopic open paths were smooth, in the sense that they had no small-scale features, one would not expect the shortest circuit to be much shorter than the innermost, since the latter encloses a smaller area.

Remark 1.2.
Between the original submission and final acceptance, we obtained results stronger than that in (1.4). First, in the arXiv note [6], we refined the strategy introduced in this paper to give the bound ESn n 2 π3(n) ≤ 1 (log n) c for any c < 1/4. Further improvements are in preparation.

Conjectures in the literature
Here we make a few brief remarks and give additional references to the literature on the subject of the chemical distance in critical percolation.
Physicists expect that there exists an exponent d min such that S n ∼ n dmin , (1.5) where the precise meaning of the equivalence ∼ remains to be specified. O. Schramm included the determination of d min in a list of open problems on conformally invariant scaling limits [28], noting that the question does not "seem accessible to SLE methods." Even the existence claim has so far not been substantiated.
Following Schramm and Kesten-Zhang, we have formulated the problem in terms of crossings of large boxes. More generally, d min is predicted to govern the chemical distance between any two points inside the same critical percolation cluster in the sense that if x, y ∈ Z 2 are connected by an open path and x − y 1 = n, then dist chemical (x, y) ∼ n dmin . (1.6) It follows from the results of Aizenman and Burchard that if x and y are at Euclidean distance of order n, then with high probability, the chemical distance between x and y is greater than n η for η > 1. One might expect, based on (1.2), that the average EJP 22 (2017), paper 75. point-to-point chemical distance can be bounded by n 2 π 3 (n), but this bound does not follow directly from the method of Morrow and Zhang. Our main result and numerical simulations suggest that a sharp upper bound would involve a quantity smaller than n 2 π 3 (n) by a power of n.
Simulations have yielded the approximation d min ≈ 1.130 . . . [8,10,32]. In contrast to other critical exponents, there is no agreement on an exact value for d min , and several proposed values seem inconsistent with each other, and with numerical results. See the introduction and bibliography in [25] for a more extensive review of these questions. In that article, the authors use the formula of V. Beffara [3] for the dimension of SLE curves d SLE(κ) = min 1 + κ 8 , 2 along with a conjectured value for d min to compare, based on simulations, the behavior of SLE(κ) with the shortest path accross a domain.

Outline of the proof
The obvious strategy is to use the following well-known fact: given any circuit L in A(n), the event that the innermost circuit γ n in A(n) (around 0) coincides with L depends only on the edges in A(n) which also lie in the interior and on γ n . One would then try to condition on the value of the innermost circuit and use RSW constructions along with independence of the region outside, to construct a large number of short paths off of L.
Unfortunately, this strategy does not work. The innermost circuit with high probability has many very irregular sections, so that it is not possible to make useful constructions with probability bounds that are independent of L.
A main contribution of this paper is to replace this approach, which attempts to use independence of the region outside the innermost circuit, with one where we instead condition on the event that an edge far from the boundary belongs to the innermost circuit, and show that with high conditional probability, there is a short path around this edge. In other words, fixing any edge e which is far from the boundary, if e ∈ γ n , we try to find in several concentric annuli around e a "detour": an open arc lying outside γ n , but with its endpoints on γ n . Given such an arc, we can form a new open circuit in A(n) by replacing a portion of γ n by the detouring open arc. Provided the resulting curve still surrounds B(n), we obtain a candidate for a circuit which could be shorter than γ n . Given > 0 fixed but small, the strategy is then to show that not only are there many detours around edges on γ n , but they can be chosen so that their length is less than times the length of the portion of γ n they detour. If most of γ n can be covered by such short detours in this manner, we can build an open circuit σ n consisting of sections of the innermost circuit, with many detours attached, and one might hope that #σ n ≤ ( + o(1))#γ n with high probability.
In trying to implement this strategy, we are faced with a number of problems: 1. We are forced to construct detours locally (that is, around edges e that are conditioned to be in γ n ), and not globally. Therefore, multiple detours around different edges might intersect. A systematic method is needed to keep track of how much of γ n we have replaced by detours.
2. We lack prior knowledge about the size of open paths in the critical cluster. It is thus not obvious that one of the many detours around each edge will have length smaller than times that of the detoured path.
3. The orientation and rough geometry of γ n could make it difficult to carry out the percolation estimates required to construct detours. Because we do not condition on the value L of γ n at any point, the global structure of γ n does not appear in EJP 22 (2017), paper 75. the detour construction. An essential point of our construction is that given the occurrence of our "detour" events, the local structure of the innermost circuit is controlled.
We address the first point by considering "shielded" detours: short detours which are also covered by a closed dual arc; see Definition 3.2. Two shielded detours are either equal or disjoint, and this allows us to estimate the total contribution of the detours to the circuit σ n .
To address the second point, we must show that very short shielded detour paths exist with positive probability in every annulus. The only tool that we have to upper bound the length of paths is the result of Morrow and Zhang, which gives asymptotics for the length of the lowest crossing (innermost circuit). We use the fact that the fractal structure of this innermost circuit of an annulus implies that it can be made much smaller than its expected size, by forcing it to lie in a very thin region. This observation, applied to outermost partial circuits within shields, allows us to construct short detours as in Definition 3.2, by constraining them to be in thin annuli. As an illustration of this idea, we give the following proposition:  for all n large enough.
Sketch of proof. We only provide an outline of the proof here. For a more detailed argument, see the proof of Lemma 5.8. The size of the lowest open crossing of [−n, n] × [−n, −(1 − α)n] is of order αn 2 π 3 (αn). Using quasimultiplicativity [23,Proposition 12.2], and the fact that the three arm-exponent is < 1, this is smaller than α 1−η n 2 π 3 (n) for some η < 1. Choosing α small enough, the result follows.
To show that most of γ n can be covered by short detours, we show in Section 6 (see equation (6.1)), that the probability of e having no shielded detour around it is small, conditioned on e lying on γ n : lim sup n→∞ P(no detour around e | e ∈ γ n ) = 0, (1.8) for edges e away from the boundary of A(n) and > 0 arbitrary.
We then estimate S n by considering separately the contributions to #σ n of the union Π of all the short detours, and the edges on γ n \Π, whereΠ is the union of the "detoured" portions of the innermost circuit: ≤ E#γ n + E#{e ∈ γ n : there is no detour around e} ≤ E#γ n + e P(no detour around e | e ∈ γ n )P(e ∈ γ n ).
Using (1.8), this gives The proof of Corollary 2.3 (see Section 2.2 for the statement) concerning the expected size of the lowest crossing is identical to the argument for the innermost circuit. To obtain the statement of convergence in probability in Corollary 2.5 (also stated in Section 2.2) we need an additional argument. Essentially, it remains to prove that the lowest EJP 22 (2017), paper 75. crossing of [−n, n] 2 cannot be smaller than o(1)EL n with positive probability. The basic idea for our proof comes from Kesten's lower bound for the number of pivotals in a box [14, (2.46)], but the requirement to find a large (of order n 2 π 3 (n)) points rather than one at each scale introduces substantial new technical difficulties. See Section 7.
For clarity, we have ignored the edges very close to the boundary in this rough sketch of our proof; for such edges, no estimate like (1.8) holds.
To obtain the estimate (1.8), we define a sequence E k (e), k ≥ 1 of events which depend on edges inside concentric annuli around e, and whose occurrence implies the existence of a shielded detour (in the sense of Definition 3.2) if e ∈ γ n . The definition and construction of E k (e) are given in Section 5, where it is also proved that uniformly in k ≥ k 0 for some c 1 > 0. A schematic representation of the event E k (e) appears in Figure 1; see also the accompanying description at the beginning of Section 5. We use closed dual circuits with defects to force the lowest crossing to traverse certain regions inside the annulus where E k (e) is defined, regardless of the "local orientation" of the innermost circuit outside. To connect the innermost circuit, the detour path and its shielding closed dual path, we use five-arm points (see Section 5.2), avoiding any conditioning on the realization of the lowest path.
To pass from (1.10) to (1.8), we show in Section 6.1 that 1. The estimate (1.10) remains true (with a different, but still n-independent constant) when we condition on e lying in the innermost circuit. See (5.24) in Section 5 and Proposition 6.1.

2.
Although the E k (e)'s are no longer independent under the conditional measure P(· | e ∈ γ n ), the dependence is weak enough to obtain an estimate on the event that none of the E k 's occur; see Proposition 6.4. Here we use arm separation tools which appeared in [7] (which we state as Lemma 6.6).

Notation and results
On the square lattice (Z 2 , E 2 ), let P be the critical bond percolation measure Given ω ∈ Ω, we say that γ is open in ω if ω(e k ) = 1 for k = 1, . . . , N . Any self-avoiding circuit γ can be viewed as a Jordan curve and therefore has an interior int γ and exterior ext γ (component of the complement that is unbounded). In this way, Z 2 is the disjoint union int γ ∪ ext γ ∪ γ. We say a self-avoiding circuit surrounds a vertex v if v ∈ int γ.
The symbols C, c will denote positive constants whose value may change between occurrences, but is independent of any parameters. Dependence on parameters is indicated by an argument, as in C(α), and we have numbered some recurring constants using subscript for clarity.

Circuits in annuli
For n ≥ 1, let B(n) be the box of side-length 2n, Let C(n) be the collection of all self-avoiding circuits in A(n) that surround the origin and, given ω, let Ξ(n) = Ξ(n)(ω) be the sub-collection of C(n) of open circuits.
We will be interested in the event which we know has 0 < inf n P(Ω n ) ≤ sup n P(Ω n ) < 1 by RSW arguments [27,30]. On Ω n we may define γ n , the innermost element of Ξ(n), as the unique γ ∈ Ξ(n) which has int γ ⊂ int σ for all σ ∈ Ξ(n). This allows us to define the random variable L n = L n (ω) = #γ n for ω ∈ Ω n 0 for ω / ∈ Ω n . This is the length of the innermost open circuit. The expected length of the innermost open circuit can be estimated using arm events.
Let A 3 (n) be the "three-arm" event that 1. The edge (0, e 1 ) is connected to ∂B(n) by two open vertex disjoint paths and 2.
In later sections, we use arm events centered at vertices other than the origin. We define them now. For v ∈ Z 2 , A 3 (v, n) denotes the event that A 3 (n) occurs in the configuration ω shifted by −v. For an edge e = (v 1 , v 2 ) ∈ E 2 , A 3 (e, n) denotes the event that 1. e is connected to ∂B(e, n) := ∂B((v 1 + v 2 )/2, n) by two disjoint open paths and 2. The dual edge e * is connected to ∂B((v 1 + v 2 )/2, n) * by a closed dual path.
In item 2, we view the boundary as a subset of R 2 and say that e * is connected to it if there is a closed dual path from e * which (when viewed as a subset of R 2 , touches it. Denoting π 3 (n) = P(A 3 (n)), we have the following simple adaptation of the result of Morrow and Zhang: Proposition 2.1. There exist C 1 , C 2 > 0 such that C 1 n 2 π 3 (n) ≤ EL n ≤ C 2 n 2 π 3 (n) for all n ≥ 1 . The characterization of the innermost circuit (based on Morrow and Zhang) we will use throughout the paper is as follows. An edge e ⊂ A(n) is in the innermost circuit if and only if the following occurs: e * is connected to B(n) * by a closed dual path, and e is in a self-avoiding open circuit surrounding B(n). One way to say this is that e has three disjoint arms (two open and one closed), with the closed arm connected to B(n) * and the open ones connecting into a circuit around B(n).
We can further define the length of the shortest open circuit. That is, set Our main result for circuits is Theorem 2.2. As n → ∞, ES n = o(n 2 π 3 (n)) .

The lowest crossing and the question of Kesten-Zhang
The proof of Theorem 2.2 applies equally well to the lengthL n of the lowest crossing of B(n). Here on H c n , where H n is the event that there is an open horizontal crossing of B(n), we defineL n = 0.    We give a proof sketch of (2.5) in Section 4. The full proof, which is somewhat involved, appears in an arXiv version of the current paper. With Lemma 2.4 in hand, we can answer the question of Kesten and Zhang.

Corollary 2.5.
Conditionally on H n , we have the convergence in probability: The proof of Corollary 2.5 will be found in Section 4.

Short detours
On the event Ω n , we will find another another circuit σ n ∈ Ξ(n) such that #σ n = o(#γ n ), whereγ n is a truncated version of the innermost circuit γ n (see equation (3.3)) in a slightly thinned version of A(n). To define this annulus, we note the following:   , there exists C 4 > 0 such that for any > 0:

Definition of shielded detours
In this section, we define the central objects of our construction, the shielded detour paths π(e), e ∈γ n .

Definition 3.2.
Given ω ∈ Ω n , ∈ (0, 1), and any e ∈γ n , we define the set S(e) of -shielded detours around e as follows. An element of S(e) is a self-avoiding open path P with vertex set w 0 , w 1 , . . . , w M such that the following hold: 3. writing Q for the subpath of γ n from w 0 to w M that contains e, the circuit Q ∪ P does not surround the origin, 4. the points w 0 + (1/2)(−e 1 + e 2 ) and w M + (1/2)(e 1 + e 2 ) are connected by a dual closed self-avoiding path R whose first and last edges are vertical (translates of {0, e 2 }) and is such that the curve formed by the union of R, the line segments from the endpoints of R to w 1 and w M −1 , and P does not enclose the origin, and 5. #P ≤ #Q. Now fix a deterministic ordering of all finite lattice paths and define π(e) to be the first element of S(e) in this ordering. If the set S(e) is empty, then we set π(e) = ∅.

Properties of the detour paths
We give the properties of the collection of detours (π(e) : e ∈γ n ) which we use in the next section to prove Theorem 2.2 and Corollary 2.5. The definition of π(e) (Definition 3.2) appeared in Section 3.1.
Let 0 < < 1. Then: 1. Each π(e) is open and for distinct e, e ∈γ n , π(e) and π(e ) are either equal or have no vertices in common.
2. If e ∈γ n and π(e) = ∅, write π(e) = {w 0 , e 0 , . . . , If e ∈γ n then the segmentπ(e) of γ n from w 0 to w M containing e (that is, the "detoured" portion of γ n ) is such thatπ(e) ∪ π(e) is a circuit that does not surround the origin. Furthermore, #π(e) ≤ #π(e) . 4. There exists C 5 ( ) > 0 such that for all n ≥ C 5 and e ∈Â(n), (The exponent 2 on on the right in (3.5) will be used in (4.4).) We must show the above properties follow from Definition 3.2. Most of the work will be in showing item 4, the probability estimate (3.5), which we will do in Section 6. Items 2 and 3 hold by definition. For item 1, we have Proposition 3.3. If ω ∈ Ω n , then for distinct e, e ∈γ n , π(e) and π(e ) are either equal or have no vertices in common.
The proof of Proposition 3.3 will be found in Section 8. For the rest of this section, we will identify paths with their edge sets. Given such detour paths π(e) (which necessarily do not share edges with γ n , due to (3.4)) and detoured pathsπ(e) (subpaths of γ n ), we construct σ n as follows. First choose a subcollection Π of {π(e) : e ∈γ n } that is maximal in the following sense: for all π(e), π(e ) ∈ Π with e = e , the pathsπ(e) andπ(e ) share no vertices and the total length of detoured paths π∈Π #π is maximal. The choice of Π can be arbitrary among all maximal ones.
We putΠ Now define σ n to be the path with edge set equal to the union of Π and those edges in γ n that are not inΠ.
We must now show that if such a construction can be made, then lim sup n E#σn ELn ≤ (so Theorem 2.2 follows). To do this we will need two tools: For ω ∈ Ω n , σ n is an open circuit in A(n) surrounding the origin.
The proofs of these two lemmas is detailed in Section 8.  Let (Π) = π∈Π #π be the total length of the detours in the collection Π. Assuming Lemmas 3.4 and 3.5, we estimate the length of σ n : We have: Since is arbitrary and EL n ≥ C 3 n 1+C4 (using (2.2) and Lemma 3.1), (2.2) gives ES n = o(n 2 π 3 (n)), finishing the proof of Theorem 2.2.
To obtain Corollary 2.5 for crossings of a box, we repeat the construction above to obtain a crossingσ n and a unionΠ of detours from the lowest crossing l n . Denoting bỹ P the probability measure conditioned on the existence of an open crossing P = P( · | H n ), +P(#{e ∈l n : π(e) = ∅} > 1/2 #l n ) By an estimate analogous to (4.1), the first probability on the right is zero. We decompose the last term further: By Markov's inequality and (4.2), the termP(#{e ∈l n : π(e) = ∅} > EL n ) is bounded by . (Recall that 0 < P(H n ) < 1 uniformly in n, soP is uniformly absolutely continuous with respect to P.) It remains to estimate the second term on the right in (4.3) and the last term on the right in (4.4). Using again that EL n ≥ C 3 n 1+C4 , we see that it will suffice to show that lim sup Recalling thatP is supported on the event {L n > 0} and P(H n ) > 0 uniformly in n, this reduces to showing lim ↓0 lim sup n→∞ P(0 <L n ≤ 1/2 EL n ) = 0. This will be done in the Section 7. In this section, we define events E k (depending on and η), which will be used in the proof of the probability estimate (3.5) (see (6.2) in Section 6.1). The formal, precise definition of E k is quite long and complicated and given in Definition 5.10, equation (5.22). The essential property of E k is Proposition 5.1. Let e be an edge of Z 2 and > 0, and let E k (e) = τ −e E k be the translation of E k by the edge e. That is, for any ω ∈ Ω, There is a constant K( ) such that the following holds. Let Proposition 5.1 will be proved in Section 5.6. As for the probability of occurrence of E k , we have the following lower bound, where A 3 (d) was defined in Section 2.1.
for some uniform constant C 7 independent of n. E k will be defined as the conjunction of a large number of crossing events (see equation (5.22)). These events will be gradually introduced over the next few subsections. Figure 1 illustrates most of the crossing events in E k .
The proof of Proposition 5.2 takes up most of the remainder of Section 5, where lower bounds for the subevents of E k are given. The proof is completed in Section 5.5.
After we prove Propositions 5.1 and 5.2, Lemmas 3.4 and 3.5, and Proposition 3.3, we can conclude Theorem 2.2.

Sketch of proof of Proposition 5.2
Because the proof of the above proposition requires many constructions, we now give a sketch of the main ideas. Let η ∈ (1, 11/10) be given and k ≥ 1. The quantity δ = η − 1 > 0 should be thought of as small. Define the annuli The event E k (e) has three main features: 1. Two closed circuits (in green in Figure 1), in Ann 1 and Ann 2 , each with two defects, in thin concentric annuli. These serve to isolate the inside of the annulus 2. An open half-circuit connected to the crossings of the annulus emanating from the origin (in red in Figure 1). This will act as a detour for the portion ofγ n inside the larger box.
In Section 5.3, we show that, given the existence of the circuits in the previous item, the open half-circuit contains at most 3 2k π 3 (3 k ) edges with high probability, where > 0 is small. 3. Boxes containing order 3 2k π 3 (3 k ) many three-arm points connected to the arms emanating from e. (The relevant connections appear in blue in Figure 1.) On E k (e) ∩ {e ∈γ n }, these points lie onγ n .
We give a lower bound of order C3 2k π 3 (3 k ) for the number of three arm points on the open arms emanating from the origin. This holds with probability bounded below independently of n, conditionally on the events in the previous items. This is done in Section 5.4.
Given these three features, the proof proceeds as follows. We first show existence of circuits with defects: we show that for any given η close enough to 1, there is a constant where X 1 (k, η) is the event that the closed circuits with defects from feature one above exist in the annuli Ann i , and the defected edges have five arms as in Figure 1. Next, we give an upper bound for the length of a thin detour. Namely, let X 2 (k, η, ) be the event that there is an open half-circuit connected to the defects from feature one, staying in Ann 2 , and having length at most 3 2k π 3 (3 k ). We show in Section 5.3 that for any > 0, there exists η( ) close enough to 1 such that for suitable values of k. Last, we show existence of many edges on paths that will function as the innermost circuit. Let X 3 (k, η, c) be the event that in boxes in the interior of the annulus, there are at least c3 2k π 3 (3 k ) edges with three disjoint arms: two open to the defects and one closed dual path to the bottom of the annulus. In Section 5.4, we prove that there exists c > 0 and a constant D 3 > 0 such that for all η close to 1 and suitable values of k, The most important thing here is that c has no dependence on η or , essentially because as η ↓ 1, the size of the boxes in which the three arm points lie does not decrease to 0. To put these pieces together, we first choose c such that (5.4) holds. Next, given > 0, choose η = η(c ) to guarantee (5.3) with c in place of . For this value of η, one also has (5.2). Combining the above three inequalities, and, for the purposes of this proof sketch, putting for suitable k. On this intersection, one is guaranteed that the volume of the detour is at most times the volume of the three-arm points in the boxes connected to the defects.
To finish the proof, one notes that E k implies that there is a three-arm connection across the annulus in which E k occurs. Thus one uses arm separation to see that if E k and A 3 (d) both occur, then one can, with uniformly positive conditional probability, route the arms from A 3 (d) to the inner and outer boundaries of the annulus to connect to the arms from E k . Thus there is a constant D 4 independent of n, η, c and k such that This step is standard, so we leave the details to the reader. Combining it with (5.5) completes the proof of Proposition 5.2.

Five-arm events and shielded circuit
In this section we put in place most of the components of the construction of the event E k . Define the boxes The relative placement of these boxes inside S(e, η3 k ) is shown in Figure 2. From this point on, we will restrict to n and k such that where K(η) is chosen so that all boxes involved have lengths at least some constant, say 10. (This includes the above boxes, but also those used in Section 5.4.) If we decrease η, then the range of valid k decreases.
The most important definition of this section is the following has an open arm γ 2 to The event M 1 is illustrated in Figure 3.
We let G 1 (k) be the event that Here and elsewhere in the paper, we will make extensive use of the following generalized FKG inequality [15,Lemma 3] (see also [23,Lemma 13]): By generalized FKG and standard gluing constructions, we have, using Proposition 5.4, for some constant C 9 independent of η. G 2 (k) is defined to be the reflection of G 1 about the vertical axis through 0. For a box B i , i = 1, . . . , 9, we let B i be its reflection about the e 2 -axis. That is, if x = (x 1 , x 2 ) ∈ Z 2 , then x ∈ B i if and only if (−x 1 , x 2 ) ∈ B i . The same applies to the "landing zones" I i . We say that G 2 (k) occurs if all the conditions in the definition of G 1 (k) occur, replacing each B i , i = 1, . . . 9, and I 1 , I 2 by B i and I 1 , I 2 , respectively. By symmetry and independence: We let R 1 (k) be the event that there is a closed left-right crossing of B 12 ; R 2 (k) is the event that there is an open left-right crossing of B 13 ; R 3 (k) is the event that there are closed left-right crossings of B 10 , B 11 and B 14 .
(whose right side depends on η, although this is not visible in the notation) implies the events: 1. There are five-arm points w and w in B 1 and B 1 , respectively.
2. There is a closed dual circuit α 1 with 2 defects near w and w in The arcα 1 of α 1 between w and w is contained in There is an open arc α 2 contained in Γ(k, η) with endpoints at w and w . Moreover, α 2 is contained in the interior of the dual circuit α 1 .
We note that there exists C 10 (η) > 0 depending on η such that P(Q 1 (k, η)) ≥ C 10 for k ≥ K(η) . In the above definition, we may choose α 1 arbitrarily, but this choice uniquely defines α 2 . We have the following Lemma 5.7. On the event Q 1 (k, η), any edge e onα 2 has 3 disjoint arms: two open and one closed. These arms reach to distance at least 3 k (η − 1)/2 from e.
Proof. If e is onα 2 , then necessarily, there is a closed path from e * toα 1 contained in the region int(α 1 ∪α 2 ). Following this closed path until we reach the closed pathα 1 , we can extend it into a closed path of at least the required length, because the path α 1 has diameter greater than 3 k . On the other hand, the pathα 2 itself has two ends, one of which necessarily has length at least 3 k /2. As for the other, it emanates from a five-arm point, itself connected to a crossing of a box of width 3 k (η − 1).
We rewrite the probability (using (5.7)) as P(e ∈α 2 , Q 1 (k, η)) P (Q 1 (k, η)) We now define crossing events in "truncated" regions, which depend on the position of e:M 3 is the event 1. There are vertical closed crossings of each component of 1)).

There are vertical open crossings of each component of
The definition of the events implies that the truncated regions considered are either rectangles, or consist of the union of two disjoint rectangles which also do not abut (see Figure 4 -this is the reason for the choice of slightly smaller rectangles with bound 3 k (2 − η)), which implies:  Then: In the second step we have used independence. Using a gluing construction and FKG, it is easy to "fill in" the truncated regions and show The point here is that the constants represented by C do not depend on η and k. This is due to the fact that the regions we must fill in have size of order 3 k (η − 1).
Summing over e in Γ(k, η), this gives The lemma now follows by Chebyshev's inequality, and the observation that for all 1 ≤ m ≤ n, one has π 3 (m, n) ≥ C 3 (n/m) −α for some C 3 > 0 and α ∈ (0, 1) (see Lemma 3.1). Combined with quasi-multiplicativity, it gives: Choosing δ sufficiently small completes the proof of Lemma 5.8.   Figure 5: The boxes appearing in the derivation of the lower bound for detoured crossings. Compare to Figure 1, where some of the macroscopic connections are shown.

Lower bound for the volume of detoured crossings
We define boxes inside the annulus B(η3 k ) \ B(3 k−1 ), and events on which these boxes will be traversed by the open arms emanating from the origin, and contribute on the order of (3 k ) 2 π 3 (3 k ) edges. The boxes are centered at the midpoints of [−3 k , −η3 k−1 ] and [η3 k−1 , 3 k ], respectively. Let The left "interior box" is Inside B 15 , we place a smaller box (a quarter of the size), also centered at x 0 (k): it is defined by The remaining boxes will serve to define crossing events to connect 3-arm points in B 16 to a closed crossing of Ann 1 (k). We let R 4 (k) be the event that there is a left-right dual closed crossing of B 22 and a top-down dual closed crossings of B 24 . Again by RSW and Harris, we obtain P(R 4 (k)) ≥ C > 0 for some constant C independent of k and η.
The important point here is that c does not depend on δ. As η ↓ 1 (see Lemma 5.8), the size of the detour shrinks, whereas the lower bound in this proposition does not change.
Proof. Recall the Paley-Zygmund inequality: if Z ≥ 0 a.s. and λ ≤ 1, then To apply this with Z = Z * 2 and P = P(· | Q 4 ), we give an estimate for the expectation and an upper bound for the second moment: Summing over e ∈ B 16 , we obtain This gives the lower bound in (5.19). We now estimate the second moment. A similar argument gives the upper bound for the first moment. For simplicity of notation, let where in the second step we have used that Q 4 is independent of the status of edges inside B 25 .
Returning to the sum, we find, for each e 1 , the bound Now we use RSW theory to rescale some of these quantities by constant factors. First, we have π 3 (d/2) ≤ Cπ 3 (d).

Definition of E k and Proof of Proposition 5.2
We can now give the definition of the events E k : Definition 5.10. Let > 0 and η = η( ) be such that (5.12) in Lemma 5.8 is satisfied.

(5.23)
To derive the lower bound Note that the definition of E k implies A 3 (3 k−1 , η3 k ) and that the connections across the annulus are easily extended.

Proof of Proposition 5.1
Proof. We will show that if the event On E k (e), there is a closed circuit C 2 with two defects near the five-arm points in B 1 and B 1 inside Ann 2 . We denote these (unique) points by x and y. Since e lies on the open, self-avoiding, circuit γ n , the latter must pass through each of the two five-arm points in the definition of M 1 , resp. M 1 , exactly once. We denote the portion of γ n between x and y, and inside C 2 , by q. We also let p =α 2 , whereα 2 is from Definition 5.6.
Claim 5.11. On E k (e) ∩ {e ∈γ n }, e * has a closed connection to the bottom of B(e, η3 k ) and to the bottom arc of the closed circuit C 2 with defects in Ann 2 .
Proof. Recall (from (5.9)) that Γ(k, The open (detour) arc σ o in Γ(k, η) between the two five-arm points and the closed arc σ c through Ann 2 \ Γ(k, η) form a circuit around e (we can connect them by two line segments of length 1/ √ 2 to make their union a closed curve). The closed arm emanating from e * reaches ∂B(e, η3 k ), so it must intersect σ c , which intersects the closed vertical crossing of B 24 in the definition of the event R 4 (k) ⊃ Q 2 . This crossing is connected to the bottom side of B(e, η3 k ).
It is important to note that since the closed arm from e * intersects the bottom of B(e, η3 k ) and e ∈γ n , the bottom part of the closed circuit in Ann 1 \ Γ(k, η) must be connected to B(n). This is what forces the "orientation" of the box B(e, d) to be such that p is indeed a detour off the innermost circuit (see Lemma 5.13 below). whose only intersection with the outermost arc p is x and y. By the previous claim, every dual edge touching q is connected to the bottom of B(e, η3 k ) by a closed dual path. This implies that q lies in the region below the Jordan curve α, which separates the box B(η3 k ) into two connected components. In particular, q is disjoint from p, except at its endpoints.
It follows that p ∪ q is a Jordan curve lying entirely inside the box B(e, η3 k ). This in turn implies Lemma 5.13. p lies outside intγ n .
Proof. The dual edge {x−(1/2)(e 1 +e 2 ), x−(1/2)(e 1 −e 2 )} crosses γ n and so one endpoint is in each component of the complement of γ n . The top endpoint can be connected to p and the bottom one can be connected to the bottom arc of C 2 , both without crossing γ n . By Claim 5.11, the bottom is in the interior of γ n , so p is in the exterior of γ n .
We can now prove Proposition 5.1, by setting K( ) = K(η). Letting P = p and Q = q, w 0 = x, and w M = y, Lemma 5.13 implies that Condition 1. in Definition 3.2 is satisfied. Condition 2. holds by the definition of the five-arm points x and y (Definition 5.3). Condition 3. follows because P ∪ Q = p ∪ q is contained in the box B(e, η3 k ), which does not contain the origin. Condition 4. follows from the existence of the closed dual arcα 1 , which is implied by the event Q 1 (5.8). Condition 5. holds because of the conjunction of Q 2 (5.11) and Q 5 (5.16) (choose · c for , where c is from the definition of Q 5 ).

Proof of inequality (3.5)
Our goal in this section is to derive the estimate (3.5). We recall it here: P(S(e) = ∅ | e ∈γ n ) < 2 (6.1) for some n ≥ C 5 . Once we show this estimate, Lemmas 3.4 and 3.5, and Proposition 3.3, we can conclude Theorem 2.2.
For k = K(η), ..., (C 4 /8) log n , we let E k (e) = τ −e E k be the "detour event" inside the annulus Ann 1 (e, k) = B(e, η3 k ) \ B(e, 3 k−1 ). (Here η is slightly bigger than 1 and K = K(η) is a constant depending on η and which is defined under (5.6).) It is defined precisely in Section 5 (see Definition 5.10). The property we need here is proved in

Conditioning on 3-arm event in a box
The next step is to replace the conditioning in (6.2) by conditioning on a "three arm" event: dist(e, ∂A(n))), (6.3) where A 3 (e, m) is the probability that e has three arms, two open and one closed, to distance m from e.
We will omit some details, since most of the arguments are lengthy but standard. To prove (6.3), we use a gluing construction that depends on the position of e inside the annulus A(n), which we split into a number of different regions: Region B is the disjoint union of four rectangles: Finally, region E is given by In each case, we use an adapted gluing construction to connect e to B(n) by a closed dual path inside an open circuit around B(n). Figure 6 depicts the partitioning we will use.  We concentrate on the proof of (6.3) in case e ∈ A. Furthermore, we assume by symmetry that e is in the top-right component of A. We only consider e ∈γ n , so d(e) = dist(e, ∂A(n)) ≥ n C4/2 . In this lemma, we define F 1 (e) to be the sure event (that is, the entire sample space) if 8d(e) > d (e) and we define F 2 (e) to be the sure event if 4d (e) > n. This is to guarantee that later in the proof, there is enough room between the boxes B(e) and H(e) (or between H(e) and K(e)) to do arm separation arguments.
Proof. If e ∈ γ n then e belongs to an open circuit surrounding B(n) in A(n). Moreover, e * is connected to ∂B(n) by a closed path contained in the interior intγ n of the circuit.
Let r 1 be the portion of the open circuitγ n obtained by traversing the circuit in one direction from e, until first time it exits H(e). Call a(e) ∈ ∂H(e) the point of exit. Let r 2 be the portion of γ n obtained by traversing the circuit in the other direction, until it first exits H(e), at a point b(e). The curve γ e contained in H(e) joining a(e) to b(e) separates H(e) into two regions, each bounded by the curve γ e and a portion of ∂H(e) \ ∂B(3n). Exactly one of these regions, R(e), say, lies inside the circuit γ n , and hence contains the portion of the closed dual path from e to ∂B(n) until it first exists H(e). Following this path from e until this exit point, we obtain a closed dual path whose endpoint c(e) must lie on ∂H(e), between a(e) and b(e). Traversing r 1 from a(e) and r 2 from b(e) toward e until the first time they enter B(e), we obtain two points a (e) and b (e) on ∂B(e) \ ∂B(3n). Following the closed path backwards similarly, we find a point c (e) lying between a (e) and b (e) on ∂B(e) \ ∂B(3n).
The proof for the paths in K(e) \ H(e) is similar.
Proposition 6.1 now follows from the next estimate: There is a constant C independent of n and e such that P(e ∈γ n ) ≥ 1 C P(F 1 (e))P(F 2 (e))P(A 3 (e, d(e))), Proof. We first introduce two events that will serve to complete a circuit around A(n), By the Russo-Seymour-Welsh theorem, the Harris inequality and independence, there is a positive C 6 independent of n such that P(C 1 ), P(C 2 ) ≥ C 6 . (6.8) To connect the partial circuits C 1 and C 2 into a circuit containing the edge e, we use a standard arms separation argument (see for example [15,Lemma 4], [23,Theorem 11]), which allows us to specify landing areas on ∂B(e), ∂H(e) and ∂K(e) for the arms in events F 1 (e) and F 2 (e), while not modifying the probability of these events by more than a constant factor. The conclusion (6.7) is then obtained using the generalized FKG inequality.
By arms separation, [23, Theorem 11], we have P(Ã 3 (e, d(e))) ≥ CP(A 3 (e, d(e))). Lemma 4]) gives the existence of a constant C independent of n, such that: P(F 1 (e)) ≥ CP(F 1 (e)) (6.11) P(F 2 (e)) ≥ CP(F 2 (e)). (6.12) It is important here that the arms in the definition of F 1 (e) and F 2 (e) appear in a definite order, as guaranteed by Lemma 6.2.  Combining (6.10), (6.11), (6.12) and (6.9), we obtain (6.7) in the case where e lies in the upper right part of the region A. Similar gluing constructions also apply in the other cases. This completes the proof of Lemma 6.3, and thus of Proposition 6.1.

Arms separation conditional on A 3
By Proposition 6.1, we have, for any e ∈Â(n): P(S(e) = ∅ | e ∈γ n ) ≤ CP(∩ C4/8 log n k=K E 2k (e) c | A 3 (e, d(e))), (6.13) where d(e) is defined in (6.4) and K = K(η) is defined below (5.6). The events E 2k depend on disjoint sets of edges, and each occurs with probability bounded below independently of k (see (5.23)), so we expect an estimate of the form (6.1).
However, we must ensure that the conditioning on the three arm event A 3 (e, d(e)) does not affect the probability of occurrence of the E 2k 's too drastically. To state our result, let d = 3 m be a (large) integer. We will later take m = log 3 d(e) . From Proposition 5.2, we have: This is the lower bound (5.24). From the definition of E 2k (Definition 5.10), we also have E 2k depends only on edges in B(3 2k+1 ) \ B(3 2k−1 ). (6.15) Set m n = (C 4 /8) log n . Our goal will be to prove that P(∩ mn k=K E c 2k | A 3 (N )) → 0 as n → ∞ (6.16) uniformly in N ≥ n C4/2 . Given this result, we find by translation invariance  We first state and prove the following claim, which will be used later in the proof of Proposition 6.4: Claim 6.5. For a sequence of integers 3 2K+1 < i 1 < i 2 < . . . < i k < . . . let B k be the event that there exists a closed dual circuit with exactly two defects (that is, two edges that are open) around 0 in Ann(i k , i k+1 ), and let k N = max{k : i k+1 < N }.
Furthermore, letB k be the event that there exists an open circuit with one defect around 0 in Ann(i k , i k+1 ). Given > 0, there is a choice of i 1 , i 2 , . . . such that i k+1 > 3 6 i k for all k and for all N .
Proof. By quasimultiplicativity, we can choose C 14 such that for all m 1 < m 2 < N , For any sequence i 1 < i 2 < . . ., let α k = P(there is an open crossing or a closed dual crossing of Ann(i k , i k+1 )).
Choose the sequence (i k ) k≥1 such that Then, estimate N )). (6.18) If A 3 (i k , i k+1 ) occurs but B k does not occur, then there must be an open crossing of Ann(i k , i k+1 ) that is disjoint from the three crossings from A 3 (i k , i k+1 ). A similar statement holds forB 1 . Therefore, by the van den Berg-Fiebig inequality [4], (6.18) is bounded by Proof of Proposition 6.4. Choose i 1 < i 2 < · · · from the previous claim corresponding to /2. For k N ≥ 3, consider the event The first step is to show that there is a constant c 1 > 0 such that for all N large, all k satisfying 3 ≤ k ≤ k N − 3, E any event depending on the state of edges in B(i k ), and F any event depending on the state of edges in B(i k+4 ) c , then On the event C N ∩ A 3 (N ), there is an innermost closed dual circuit with two defects in each annulus Ann(i k , i k+1 ). (This circuit is also vertex self-avoiding.) For a given dual circuit C in Ann(i k , i k+1 ) with edges e 1 , e 2 on C, we let Circ k (C) be the event that C is the innermost closed dual circuit with defects e i . Generally, if A 3 (N ) does not occur, then the event Circ k (C) means that C is closed, e 1 and e 2 are open, and there is no closed circuit with two defects around 0 in Ann(i k , i k+1 ) entirely contained in the union of C with its interior.
Then we can decompose To decouple, we must introduce three events that build A 3 (N ). Every closed dual circuit C or D above contains two arcs between its defects. Given a deterministic ordering of all arcs and a dual circuit C, let A i (C) be the i-th arc of C in this ordering. For i = 1, 2, let X − (C, i) be the event that 0 is connected to e 1 and e 2 by disjoint open paths in the interior of C, and to A i (C) by one closed dual path in the interior of C. Let C k− N be the × P(X + (D, j), F, C k+ N ) .
The effect of this decoupling will be to "reset" the system outside of the outer circuit D, so that the event E no longer significantly affects the occurrence of F . Intuitively speaking, E could affect the system by biasing certain circuits C to appear in Ann(i k , i k+1 ), and these could change the conditional probability of F . However, a lemma from [7, Lemma 6.1] below will show that the second circuit D will mostly remove this possible bias and allow the system to start fresh. We give here a modification of that lemma. We omit the proof, which uses extensions of arm separation techniques developed by Kesten and is essentially the same proof as that found in [7]: Lemma 6.6. Consider dual circuits C in Ann(i k , i k+1 ), D in Ann(i k+3 , i k+4 ), edges e 1 , e 2 on C and f 1 , f 2 on D respectively. For i, j = 1, 2, let P (C, D, i, j) be the probability, conditional on the event that all edges in C \ {e 1 , e 2 } are closed and e 1 , e 2 are open, that (1) there are disjoint open paths from e i to f i in the region between C and D (not including C), (2) there is a closed dual path from A i (C) to A j (D) in the region between C and D (not including C), (3) D is the innermost closed dual circuit with defects f 1 , f 2 around 0 in Ann(i k+3 , i k+4 ), and (4) B i k+1 ∩ B i k+2 occurs. We similarly define C , D , i , j , etc. There exists a finite constant C 8 (it does not depend on the particular choice of circuits, defects, or i, i , j, j ) such that To apply the lemma, we rewrite the probability in (6.20), (6.21) as: Similarly, Multiplying these and using Lemma 6.6, one obtains Dividing gives This shows (6.19) with To finish the proof of (6.16), we use estimate (6.19) to show that at least one E 2k occurs. The idea is to consider a maximal sub collection F 1 , F 2 , . . . of the E 2k 's such that F 1 depends on the state of edges in B(i 3 ), F 2 depends on the state of edges in B(i 8 ) \ B(i 7 ), F 3 depends on the state of edges in B(i 13 ) \ B(i 12 ), and so on. Write r n for the largest k such that F k depends on edges in B(3 2mn+1 ), where m n was defined below (6.15). Then ≤ (1 − c 1 · a n,N ) rn , where a n,N = min However by the bound (6.14) and Claim 6.5, one has Combining this with Claim 6.5, one has for n large and uniformly in N ≥ n C4/2 .

The lower tail ofL n
In this section we give a proof sketch of the following result: lim sup n P(0 <L n < n 2 π 3 (n)) = 0. (7.1) To limit the length of the current paper, we only provide a sketch of Lemma 7.1. The full proof can be found in the online version of the paper [5].
The idea of the proof is similar to that of Kesten's proof that at criticality, the expected number of edges which are pivotal for a crossing event is at least order log n [14]. We . Calling e k the edge at the intersection of p and T k , we then use independence of the edge variables for edges in the region below and to the right of e k to build many "three-arm" edges in this region in annuli centered at e k . The crucial point is that each such edge will have two disjoint open arms to T k and one closed arm to p, and will therefore be an edge on the lowest crossing of the box. Since there is a lower bound for the probability of many such edges existing in each annulus, we obtain that with high probability, many such edges exist in at least one annulus, and this impliesL n ≥ n 2 π 3 (n) with high probability.
The main difficulty in our construction (and it is this point that makes ours more complicated than the one in Kesten's proof) is that we do not want only log n number of such edges on the lowest crossing, but at least n 2 π 3 (n). This corresponds to the fact that in each annulus, Kesten needs only to produce one pivotal point, whereas we need to build many. For us to do so, the region below and to the right of e k must have many open spaces. Specifically, we must first show that with probability of order 1 − o (1), in each large enough annulus centered at e k , we can find a box of size at least δ n, for some c and δ > 0, which lies entirely in the region below and to the right of e k .

Sketch of the proof of Lemma 7.1
Let A n = A n ( ) be the event in the probability in (7.1). Let M K = M K (n) be the event that there are at most K disjoint open crossings of the box [−n, n] 2 . Note that by the BK-Reimer inequality and the RSW theorem, uniformly in n.
We further let D k = M k \ M k+1 be the event that the maximal number of disjoint crossings equals k. Then if n is large. Here, C , c > 0 are independent of , k and n. It then follows that Letting K = log log 1 , we obtain (7.1).
To derive (7.2), we condition successively on top-most paths T 1 , . . . , T k . T 1 is defined as the horizontal open crossing of B(n) such that the region above T 1 is minimal. T 2 is then defined as the highest crossing of the region below T 1 , and T i , i = 3, . . . , k is defined analogously.
For any k-tuple of paths t 1 , . . . , t k that is admissible in the sense that is independent of the status of edges below t k (see [13,Prop. 2.3]). Moreover, the event where R(t k ) is the event that there is a dual closed path from e * , where e is some edge on the path t k , to the bottom of the box [−n, n] 2 . Note in particular that R(t k ) is independent of the status of edges on and above t k .
On E(t 1 , . . . , t k ), there is a unique left-most closed dual path from t k to the bottom of [−n, n] 2 , and we denote it by P (t k ). It is characterized by the following three-arm condition: each dual edge on p has one closed dual arm to the path t k , a disjoint closed dual arm to the bottom of the box, and an open arm to the left side of the box in the region below t k . Given a closed dual path p in the region below t k , the event E(t 1 , . . . , t k ) ∩ {P (t k ) = p} is independent of the status of edges in the region below t k and to the right of p. Our goal will be to use this independence to connect at least √ n 2 π 3 (n) points to T k by two disjoint open paths and to P (T k ) by a closed dual path. On D k , we can uniquely define the edge e k = {x k , y k } where P (T k ) meets T k .
Let α(T k ) and β(T k ) be the left and right endpoints, respectively, of T k . By RSW, we have, for some c = c(ν) > 0, n large, small, and all k, Also by RSW, we can arrange that T k remains at distance 10 ν n from the each of the 4 corners: for n large, small, and all k,  On D k , let I k = I k (n) be the event e k is close to the endpoints of T k and the corners or [−n, n] 2 and that dist(e k , α(T k )) ≥ 2 ν n, (7.3) dist(e k , β(T k )) ≥ 2 ν n, (7.4) dist((T k ∪ P (T k )), Corner i ) ≥ 10 ν n, i = 1, . . . , 4, (7.5) and there is a pair u, v ∈ Z 2 such that u ∈ T k , v ∈ P (T k ), dist(u, e k ) ≥ 10 ν n, dist(v, e k ) ≥ 10 ν n (7.6) and dist(u, v) < δ n. (7.7) Using the fact that the event that two points u ∈ T k and v ∈ P (T k ) coming too close would generate a six-arm event (see Figure 8), and the six arm exponent is greater than 2, we show the following: Lemma 7.2. There exist C, η > 0 and 0 < δ < ν < 1/2 such that for small enough > 0, It follows that to estimate P(A n , D k ), we can write P(A n , D k ) ≤ 10e c + C log 1 k c + P(D k , I k ) + t1,...,t k ,p P(A n , D k , T i = t i ∀i, P (T k ) = p), (7.8) where the sum is only over t i and p such that I k occurs and EJP 22 (2017), paper 75.
1. dist((T k ∪ P (T k )), Corner i ) > 10 ν n, for all i = 1, . . . , 4 and 2. e k is at least distance 2 ν n from the bottom and right sides of [−n, n] 2 , where e k is the edge where p and t k meet. Condition 2 follows from an argument similar to the one given to estimate P(I k ), involving the half-plane 3-arm argument: this is where the term C log 1 k c in (7.8) comes from.

Consider concentric annuli
Ann(e k , 2 l ) = B(e k , 2 l ) \ B(e k , 2 l−1 ), for l = 4 + log 2 δ n , . . . , log 2 ν n . On {D k , T i = t i ∀i, P (T k ) = p}, if Ann(e k , 2 l ) contains more than n 2 π 3 (n) points connected to t k by two disjoint open paths, and to p by a closed dual path, thenL n ≥ n 2 π 3 (n). We claim that with probability bounded away from 0 independently of l, n, t k and p, the number of such edges in Ann(e k , 2 l ) is bounded below by C 2δ n 2 π 3 ( δ n). (7.9) Choosing δ appropriately, this is bounded below by n 2 π 3 (n) for small.
To obtain the lower bound (7.9) with uniformly positive probability, we use the second moment method to find a large number of three arm points in a box inside the region R below t k and to the right of p. To this effect, we need to show that it is always possible to find such a box of side-length r at least δ n. In addition, to use RSW and connect the three arm points to p and t k , we need the box to be at a distance from these crossings that is roughly comparable to r, and the crossings themselves to be separated on this scale. It turns out that a suitable box can always be found, regardless of the choice of t 1 , . . . , t k , p, as long as I k holds, and the conditions 1. and 2. are satisfied, but showing this requires a somewhat delicate geometric argument. We refer the reader to the full proof in [5] for details. Using this construction in the order log 1 annuli Ann(e k , 2 l ), we obtain the estimate P(A n , D k , T 1 = t 1 , . . . , T k = t k , P (T k ) = p) ≤ (1 − C) c log 2 1 P(T 1 = t 1 , . . . , T k = t k , P (T k ) = p), which, together with (7.8) and Lemma 7.2 implies the desired result (7.2).

Lemmas of a topological nature
In this section, we complete the detour construction by providing proofs of the proposition and lemmas assumed in Sections 3 and 3.1.
First, recall Proposition 3.3: If ω ∈ C n , then for distinct edges e, f ∈ γ n , π(e) and π(f ) are either equal or have no vertices in common.

Proof of Proposition 3.3
Because π(e) was defined as the first element of S(e) in a deterministic ordering, we see that it will suffice to show: given P (e) ∈ S(e) and P (f ) ∈ S(f ), if V (P (e)) ∩ V (P (f )) = ∅, (their vertex sets intersect) then P (e) ∈ S(f ) and P (f ) ∈ S(e). (8.2) Now, suppose the intersection of π(e) and π(f ) is non-empty. Then π(e) ∈ S(e) ∩ S(f ), so π(e) = π(f ).
Let P (e) ∈ S(e). By extending the two open ends of the closed path in condition 4. to meet the midpoints of the edges {w 0 , w 0 − e 1 } and {w M , w M + e 1 }, we can form a Jordan curve θ(e) (i.e. a continuous, self-avoiding closed curve) by traversing the closed dual path from w 0 + (1/2)(−e 1 + e 2 ) to w M + (1/2)(e 1 + e 2 ), traversing the path Q(e) (listed as Q in the definition of S(e)), and returning to w 0 + (1/2)(−e 1 + e 2 ). By the Jordan Curve Theorem, any connected set of which does not intersect θ(e) must lie completely on either side of θ(e).
We apply the preceding to P (e) \ {w 0 (e), w M (e)} and P (f ) \ {w 0 (f ), w M (f )}. By assumption (Condition 1.) both sets lie in the exterior of γ n , so neither can intersect Q(e) or Q(f ). Since they consist of open edges, they also cannot intersect either the dual portion of θ(e) or that of θ(f ). Thus, assuming (8.1), we have Claim 8.1. Except for the endpoints w 0 (e), w 0 (f ), w M (e), w M (f ), the paths P (e) and P (f ), considered as the union of their edges and vertices, lie in the same connected components of R 2 \ θ(e) and R 2 \ θ(f ).
We now assert that this implies: Claim 8.2. The endpoints of P (e) and P (f ) coincide.
To see this, we start with the following Proof. The vertices w 1 (e) and w 1 (e) − e 1 lie on opposite sides of θ(e), since they can be connected by a segment which intersects the curve exactly once. The vertex w 0 (e) − e 1 is in the same component as w 1 (e) − e 1 , so it is in the component which does not contain P (e). However each vertex of γ n \Q(e) can be connected to w 0 (e)−e 1 by simply following γ n (without crossing θ(e)). Claim 8.3 implies that P (f ), lying as it does in the same component as P (e), must have both of its endpoints in Q(e) (since they must be in γ n ). The argument is symmetric, so that both endpoints of P (e) must lie in Q(f ), so that w 0 (e) = w 0 (f ) and w M (e) = w M (f ).
At this point, we have established Claim 8.2.
The coincidence of the endpoints of P (e) and P (f ) implies also Q(e) = Q(f ), so that P (e), together with the dual path from Condition 4. in the definition of S(e), satisfies Conditions 1-5 defining S(f ). We have proved Proposition 4.

Proof of Lemma 3.4
By Proposition 3.3, the set E(Π) is a disjoint union of paths P (e) = π(e) for a finite collection of edges e ∈ σ n . Our strategy will be to inductively replace each portion Q(e) of γ n by the detour path P (e) and show that at each stage, we still have a circuit around the origin. In other words, enumerating the paths P 1 , P 2 , . . ., and Q 1 , Q 2 , . . ., we replace Q 1 with P 1 to create γ (1) n from the original circuit γ n . Then we replace a portion of γ (1) n (which is Q 2 ) with P 2 to create γ (2) n , and so on.
Note that at stage k, the path P k+1 satisfies the definition of -shielded detour with γ n replaced by γ (k) n . Indeed, since all paths Q are disjoint for paths in Π, points 2-5 are obvious. Furthermore, assuming that P k+1 satisfies point 1 with γ n equal to γ (k−1) n , it must also satisfy it with γ n equal to γ n since the paths Q i are disjoint (choose a point near an endpoint of Q k+1 ), so this implies that P k+1 must cross from the interior to the exterior. It cannot cross γ n , so it must cross one of the P i 's for i = k + 1. This is a contradiction, since the P i 's are disjoint.
Therefore to prove Lemma 3.4, it will suffice to show the following: Claim 8.4. Let σ be a self-avoiding circuit surrounding the origin, and let P denote a self-avoiding path with endpoints in σ, such that P satisfies Conditions 1-5 in the definition of π(e), with γ n replaced by σ. Then (σ \ Q) ∪ P is a circuit in A(n) surrounding the origin. Here Q is defined relative to σ and P as in Condition 3. in the definition of π(e). Q o ⊂ int(P ∪ (γ n \ Q)).
The connected set Q o lies either in int(P ∪ (γ n \ Q)) or in ext(P ∪ (γ n \ Q)). It will suffice to exclude the latter case. For this, we will use a simple intermediate result. Proof. Any continuous path from a point in int(B ∪ C) to infinity must cross either B or C. If it does not cross B, it must cross C at a point in int(A ∪ B), after which it must cross A, and the lemma follows.
Let us now return to the proof of Lemma 3.4. We assume Q o ⊂ ext(P ∪ (γ n \ Q)).
EJP 22 (2017), paper 75. Now, assume (8.5) holds. P is contained the boundary of both int(P ∪ Q) and int(P ∪ (γ n \ Q)). Recalling that we have also (8.3), we see that these two Jordan domains are disjoint. Thus, every point in a sufficiently small neighborhood of a point of P o lies in exactly one of these two domains (if it is not in P ). It follows that any curve joining a point in P o to infinity, intersecting P o only at its starting point, must intersect either γ n \ Q or Q, so P o ⊂ intγ n , a contradiction to Condition 2. in the definition of π(e). At this point, we have established Claim 8.5. Applying Lemma 8.6, find 0 ∈ intγ n ⊂ int(P ∪ (γ n \ Q)), which is Claim 8.4.

Proof of Lemma 3.5
By Proposition 3.3, the set E(Π) is a disjoint union of sets of the form π(e), where e ranges over a finite collection S of edges inΠ. If e ∈γ n \Π and π(e) = ∅ then, again by This is because if the intersection is non-empty, thenπ(e) andπ(f ) must share a common segment, which forces π(e) ∩ π(f ) = ∅. (8.6) Indeed, ifπ(e) andπ(f ) coincide, then the two initial and two final vertices of π(e) and π(f ) coincide by Condition 2. in the definition of S(e), and otherwise one endpoint ofπ(e) must be equal to some non-endpoint vertex ofπ(f ). By the Jordan Curve Theorem, π(e) must then intersect π(f ). Given (8.6), Proposition 3.3 now implies π(e) = π(f ), which contradicts the assumption on e.