Narrows and obstacles

The present subsection and the next will use the infrastructure developed in§3.1 to prove some bounds on the probabilities that contour lines cross certain regions in specified ways. This is roughly in the spirit of the Russo–Seymour–Welsh theorem for percolation, though the proofs are entirely different.

The following lemma is an estimate for having a crossing by TG^∗-hexagons wherehis negative between two arcs on the boundary of some subset of the domain, conditioned on some zero-height interface paths. The statement below is slightly complicated, because we need to keep the geometric assumptions quite general. In percolation, boundary values do not play a role, of course. But in our case we need the crossing estimate in the case where one boundary arc of the domain is conditioned to be an interface.

Lemma 3.8. (Narrows) For every ε>0 there is a δ=δ( ¯Λ, ε)>0 such that the fol-lowing crossing estimate holds. Assume (h) and (D). Let K be the event that a fixed collection {γ1, ..., γk} of oriented paths in TG^∗ are contained in oriented zero-height in-terfaces of h, and suppose that P[K]>0. Let α⊂D\(γ1∪...∪γk) be a simple path that has both its endpoints on the right-hand side (positive side)of γ1. Let A be the domain bounded by α and a subarc of γ₁, and assume that A does not meet the left side of γ₁ and that A∩(γ¯ 2∪γ3∪...∪γk∪∂D)=∅. Let α₁, α₂ and α⁰ be three disjoint subarcs of α, where α₁ contains one endpoint of α, and α₂ contains the other endpoint of α. (See Figure 3.1.) Suppose that each point in α⁰ is contained in a TG^∗-hexagon whose center is outside A. Set d₁:=sup_z∈α0dist(z, γ₁). Let Cbe the event that there is a path crossing from α₁ toα₂ inAinside hexagons where his negative. Let d^∗be the infimum diameter of any path connecting α⁰ to γ₁∪... γ_k∪∂Dwhich does not contain a subpath connecting α\(α₁∪α₂)to γ₁ in A. If

d₁+16δmin{d^∗,dist(α₁, α⁰),dist(α₂, α⁰),diamα⁰}, (3.24) then

P[C | K]< ε.

The idea of the proof is to observe the effect that such a crossing would have on certain averages of heights of vertices, and thereby conclude that it is unlikely. The challenge in the implementation of this strategy is to condition on a crossing in such a way that the expected heights are easy to estimate.

Proof. SetN:=₁

2diamα⁰

. Assume that (3.24) holds. Note that diamγ₁>diamα⁰−2d₁.

A

α1

α2

α⁰

γ1

Figure 3.1. Setup in the narrows lemma.

Choose some point z0∈α⁰. For j=1, ..., N let zj be a point in α⁰ at distance j from z0. Now for each zj we letsj be some center of a hexagon that containszj satisfying sj∈A. Then/ |sj−sj⁰|>|j−j⁰|−O(1). LetU be the union of V∂ and the set of vertices adjacent to any one of the paths γ1, ..., γk. (These are precisely the vertices v where h takes boundary values, or the sign ofh(v) is determined byK.) Set

X:= 1 N

N

X

j=1

h(sj),

b:=E[X|K] and fix someε⁰>0. We first claim that

E[(X−b)²| K]< ε⁰ (3.25)

ifδ=δ(ε⁰,Λ)>0 is sufficiently small. Let¯ h_U denote the discrete-harmonic extension of the restriction ofhtoU and setXU:=N⁻¹PN

j=1hU(sj). Note thatE[X−XU|K, hU]=0, and henceE[XU|K]=b. For eachu∈U andj∈{1, ..., N}letp(j, u) denote the probability that a simple random walk started fromsj first hitsU atu. Also setp(u):=N⁻¹PN

j=1p(j, u).

ThenXU=P

u∈Up(u)h(u). Consequently, E[(XU−b)²| K] = X

u,u⁰∈U

p(u)p(u⁰)(E[h(u)h(u⁰)| K]−E[h(u)| K]E[h(u⁰)| K]).

LetZ(u, u⁰) denote the term in parentheses corresponding to the summand involving u andu⁰. ThenE[(X_U−b)²|K] is just the average of the conditioned covariancesZ(u, u⁰)

weighted byp(u)p(u⁰). We know from (3.1) thatE[h(u)²|K] is bounded by a constant depending only on ¯Λ. It then follows by the Cauchy–Schwarz inequality that the same is true for Z(u, u⁰). Consequently, to prove thatE[(XU−b)²|K] is small, it suffices to show that when (u, u⁰) is chosen with probabilityp(u)p(u⁰) it is very likely that|Z(u, u⁰)|

is small. Suppose that we select j from {1, ..., N} uniformly at random and given j selectu∈U with probabilityp(j, u). Independently, we also select (j⁰, u⁰) with the same distribution. It suffices to show that|Z(u, u⁰)|is likely to be small, and by Corollary 3.5 it suffices to show that the distance betweenuandu⁰ is likely to be large. Since|sj−sj⁰|=

|j−j⁰|+O(1) is unlikely to be much smaller than diamα⁰, which is larger thanδ⁻¹(d1+1), it follows from Lemma 2.1 (hit near) that for any fixedRthe probability that|u−u⁰|<R tends to zero asδ!0. Consequently,

E[(X_U−b)²| K]<¹₂ε⁰, provided thatδis sufficiently small.

SetXj:=h(sj)−hU(sj). Recall from§2.1 that, given the restriction of hto U, the function h−hU is the DGFF on VD\U with zero boundary values on U. Therefore, by (2.5),E[XiXj|K, hU]=¹₆G(si, sj), whereG(v, u) is the expected number of visits tou by a random walker started atvand stopped when it hitsU. From Lemmas 2.1 and 2.3,

G(sj, sj⁰)6

O(1) log(d1/(|sj−sj⁰|∨1)), if|sj−sj⁰|<¹₂d1, O(1)(d1/(|sj−sj⁰|∨1))^ζ1, if|sj−sj⁰|>¹2d1. Since|sj−sj⁰|>|j−j⁰|−O(1) andζ1∈(0,1), these estimates give

E[(X−XU)²| K] = 1 N²

N

X

j,j⁰=1

G(s_j, s_j0) 6 =O(1)

d₁ N

ζ1

.

Now (3.25) follows for sufficiently small δ=δ(ε⁰,Λ)>0, since¯ X−XU is independent of XU and they are also independent given K.

We now claim that b>0 if δ is sufficiently small. Let c be the constant given by Lemma 3.1. If u is a fixed vertex adjacent to γ₁ on the right, then E[h(u)|K]>1/c, by (3.2). On the other hand,E[|h(u)||K]<cfor everyu∈U. By (3.24) and Lemma 2.1, it follows that when δ is small with high probability a random walk starting at any s_j is likely to first hitU at a vertex adjacent to the right-hand side ofγ₁. Thus, when δis small, we haveb=E[X_U|K]>0. Also, clearly,b6c.

Now seta=b+1. Let Qdenote the union of the closed hexagons in TG^∗ for which h(H)∈[0, a] and let Q⁰ denote the union of the edges in TG^∗ that are on the common boundary of two hexagonsH₁ andH₂satisfyingh(H₁)<0 anda<h(H₂). LetQ₀ denote

1 2 3

4,6 7

5 8

9 10 11 12

Q0

A⁰

Figure 3.2. A portion of the sequence of hexagons adjacent to∂A⁰\∂A.

the connected component of (Q∪Q⁰)∩A¯ that contains γ1∩∂A. LetQ denote the event Q0∩(α\(α1∪α2))=∅. IfC holds, then the corresponding crossing by hexagons where h is negative separatesγ1∩∂Afrom α\(α1∪α2) in A, and hence Qholds as well. Thus, C ⊂Q.

On the eventQ, letA⁰ be the connected component of ¯A\Q0that containsα⁰, and letUQdenote the set of centers of hexagonsHsuch thatH∩A⁰∩Q06=∅. Clearly,h(v)<0 orh(v)>afor eachv∈UQ. SinceA⁰andQ0are connected, it is immediate to verify (using the Jordan planar curve theorem) thatA⁰ is simply connected and∂A⁰\∂A⊂Q0 is con-nected. The closed hexagons of TG^∗ with centers inUQ form a sequence (possibly with repetitions) with each pairH, H⁰ of consecutive hexagons along the sequence satisfying H∩H⁰\Q06=∅andH∩H⁰∩Q06=∅. (See Figure 3.2.) Ifv, u∈UQ are centers of consecu-tive hexagons in this sequence, then it is impossible thath(v)<0 andh(u)>a(otherwise, the boundary between the hexagons would be in Q⁰). Thus, either h(UQ)⊂(a,∞), or h(UQ)⊂(−∞,0). LetQ+ be the event thatQoccurs and minh(UQ)>a, and letQ− be the event thatQoccurs and maxh(UQ)<0.

We now want to estimateE[X|K,Q−] andE[X|K,Q+]. LetU⁰be the set of vertices that are either in hexagons adjacent to Q₀ or in U. Sincea=OΛ¯(1), it is clear that the proof of (3.2) givesE[|h(u)||K, U⁰,Q±]=OΛ¯(1) foru∈U⁰. On the other hand, Lemma 2.1 shows that at least 1−O(δ^ζ1) of the discrete-harmonic measure onU⁰starting from every s_j is in U_Q. If Q− holds and u∈U_Q, then E[h(u)|K,Q−, U_Q] is negative and bounded away from zero, by the corresponding analog of the left-hand side of (3.2). Thus, we find that E[X|K,Q−] is negative and bounded away from zero whenδ=δ(ε⁰,Λ)>0 is small.¯

Since E[(X−b)²|K]<ε⁰ and b>0, we conclude that by choosing ε⁰>0 small it can be guaranteed thatP[Q−|K]<¹₂ε.

OnQ+, we clearly have h(u)>a>b+1 on every u∈UQ. Thus, as above, it follows that whenδis smallE[X|K,Q+]>b+¹₂. This again implies thatP[Q+|K] can be made smaller than ¹₂ε. SinceC ⊂Q+∪Q−, this completes the proof.

Next, we formulate an analogous lemma for crossings near the boundary of the domain.

Lemma 3.9. (Domain boundary narrows) There is a constant Λ0=Λ0( ¯Λ)>0 such that for every ε>0 there is a δ=δ( ¯Λ, ε)>0 such that the following crossing estimate holds. Assume (h) and (D), assume that ∂+ is a simple path contained in ∂D and that h∂>−Λ0 on ∂+∩V∂. Set ∂−:=∂D\∂+. Let K be the event that a fixed collection {γ1, ..., γk} of oriented paths in TG^∗ are contained in oriented zero-height interfaces of h, and suppose that P[K]>0. Let α⊂D\(γ1∪...∪γk) be a simple path that has both its endpoints on ∂+. Let A be the domain bounded by αand a subarc of ∂+, and assume that A∩(γ¯ 1∪γ2∪...∪γk∪∂−)=∅. Let α⁰⊂αbe a subarc. Suppose that each point in α⁰ is contained in a hexagon whose center is outside A. Set d₁:=sup_z∈α0dist(z, ∂+). Let α₁ be a subarc of αthat contains one of the endpoints of α, and let α₂ be a subarc of αthat contains the other endpoint of α. Let C be the event that there is a path crossing from α₁ toα₂ inAinside hexagons where his negative. Let d^∗be the infimum diameter of any path connecting α⁰ to γ₁∪... γ_k∪∂Dwhich does not contain a subpath connecting α\(α₁∪α₂)to ∂+in A. If

d1+16δmin

d^∗,dist(α1, α⁰),dist(α2, α⁰),diamα⁰ , (3.26) then

P[C | K]< ε.

Proof. The proof is slightly simpler but essentially the same as that of Lemma 3.8 (narrows). We use the same notation as in that lemma, and only indicate the few differences in the proof. In the present settingb=E[X|K] can be made larger than−2Λ0

by takingδ >0 small. Here, we defineQ₀as the connected component of (Q∪Q⁰∪∂+)∩A¯ that contains∂+∩∂A. Observe thatU_Q∩∂D=∅onQ−∩K. It follows that E[X|K,Q−] is negative and bounded away from zero (by a function of ¯Λ) when δ >0 is small. By taking Λ₀>0 sufficiently small, we can make sure that E[X|K,Q−]<−3Λ₀. But since E[(X−b)²|K] is arbitrarily small and b>−2Λ₀, this makes P[Q−|K] small. The rest of the argument is essentially the same.

The previous lemmas will help us control the behavior of the continuation of contours near existing contours or the boundary of the domain. The next lemma will help us control the behavior in the interior away from existing contours.

Lemma 3.10. (Obstacle) For every ε>0 there is some constant c=c(ε,Λ)>0¯ such that the following estimate holds. Assume (h)and (D). Let K be the event that a fixed collection {γ1, ..., γk} of oriented paths in TG^∗ are contained in oriented zero-height interfaces of h,and suppose that P[K]>0. Let U be the union of V∂ and the vertices of TG adjacent to γ:=γ1∪...∪γk. Let g be a function defined on the vertices of TG that is 0 on U. Let bγg denote the union of the interfaces of h+g that contain any one of the paths γ1, ..., γk. Let B(z0, r)be a disk of radius r that is centered at some vertex z0

satisfying |g(z0)|>¹2kgk∞. Let d>0 and suppose that at distance at most ε⁻¹dfrom z0

there is a connected component of γ∪∂D whose diameter is at least εd. Also assume that kgk∞/k∇gk∞>cr>c. Then

P[bγg∩B(z0, r)6=∅| K]6ckgk⁻²_∞ logd r.

Proof. With no loss of generality, we assume that g(z0)>0. Set q:=kgk_∞/k∇gk_∞ andr1:=₁₀¹q. Since between any two verticeszandz⁰in TG there is a path in TG whose length is at most 2|z−z⁰|,

min{g(z) :z∈B(z0, r1)}>g(z0)−2r1k∇gk∞=g(z0)−¹₅kgk∞>¹4kgk∞.

As g=0 on U, it follows thatε⁻¹d>r₁. Since we are assuming thatq >cr, and we may assume thatc is a large constant which may depend on ε, it follows thatd/r>100, say.

Thus, we also assume, with no loss of generality, that kgk∞>√

c, since the required inequality is trivial otherwise.

LetX denote the average value ofhon the vertices inB(z₀, r). The inequality (3.5) andd/r>100 give

E[X²| K]6O_ε,Λ¯(1) logd

r. (3.27)

If γ₁ is not a closed path, we start exploring the interface of h+g containing γ₁ starting from one of the endpoints of γ₁ until that interface is completed orB(z₀, r) is hit, whichever occurs first. (This may entail going through several of the interfaces γj, j >1.) If that interface is completed before we hit B(z0, r), we continue and explore the interface of h+g containingγ2, and so forth, until finally either all ofγbg is explored or B(z0, r) is hit. Let Q denote the event thatB(z0, r) is hit, and letβ be the interfaces explored up to the time when the exploration terminates.

Let U⁰ be the union of U with the vertices adjacent toβ. Since we are assuming thatq >cr,r>1 andkgk_∞>√

c, and since cmay be chosen arbitrarily large, Lemma 3.2 shows that for every vertexv∈U⁰, we have that

E[|h(v)+g(v)| | K,Q, β]6kgk∞

100 . (3.28)

(Note that conditioning on Q and β amounts to conditioning that h+g >0 on vertices adjacent to the right-hand side of β and h+g <0 on vertices adjacent to the left-hand side. Consequently, the lemma applies.)

Now letxbe any vertex in B(z0, r). For each u∈U⁰, let pu denote the probability that a simple random walk started atxwill first hitU⁰inu; that is, the discrete-harmonic measure fromx. Then (3.28) gives

Since we are assuming thatq>cr, we may assume that the right-hand side is less than₁₀¹. Recall thatg>¹4kgk∞insideB(z0, r1). OutsideB(z0, r1), the trivial estimateg>−kgk∞

applies. When these estimates are applied to (3.29), one gets E[h(x)| K,Q, β]−kgk∞

We may take expectation with respect toβ and average with respect to xto conclude thatE[X|K,Q]6−¹₉kgk∞, which implies, by Jensen’s inequality, that the lemma now follows from the above and (3.27).