Bounds on the two-arms probability for percolation on Z^d

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University of Amsterdam

Korteweg de Vries Institute for Mathematics

Bounds on the two-arms

probability for percolation on Z

d

Master of Science in

Stochastics and Financial Mathematics

Master Thesis

Diederik Gerrit Pieter van Engelenburg

Supervisor

Prof. Dr. J. van den Berg (VU) Second reader

Dr. S.G. Cox

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Abstract

Consider standard site percolation on the d-dimensional lattice. Raphael Cerf [Ann. of Prob. 43. (2015)] found a new polynomial upper bound for the two-arms probability in all dimensions d ≥ 2, which holds at criticality. We prove that it holds uniformly in all p. Next, we prove a polynomial lower bound for the two-arms probability using static renormalization arguments. This result is only valid at criticality, as the event decays to zero much faster than polynomial below and above the critical point. We finally show, using the same arguments, that if the two-arms exponent is bigger than 2d2+ 2d − 2, there is no percolation at criticality.

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Introduction

We consider percolation on the d-dimensional (hyper)cubic lattice Ld = (Zd_{, E}d_{), where we say that a site is open with probability p and closed with}

probability 1−p, independent of all other sites. The model is introduced rig-orously in Chapter 2 together with other preliminary results. The research on percolation splits into three rather different sub-area’s: two-dimensions, high dimensions and the intermediate dimensions. High dimensions should be thought of as d ≥ 7. We focus on dimension three (and the other inter-mediate dimensions).

Many results for p < pc and p > pc hold in all dimensions, such as the

uniqueness of the infinite open cluster and exponential decay of the one-arm event below criticality. Percolation at the critical point itself is better understood in two and high dimensions. Two dimensions is rather special as we can use many planar arguments. In high dimensions, we have a different toolbox and this is related to ‘mean field’ behavior. The intermediate case is different as both toolboxes do not seem to provide any help when p = pc.

A remarkable result by Grimmett and Marstrand [16] and Barsky, Grim-mett and Newman [3] that does hold in all dimensions d ≥ 2 states: there is no percolation at the critical point in the half space. To prove this, they used two different renormalization arguments, but thus far no one has been able to generalize these arguments to percolation in the full space.

Raphael Cerf [7] pointed out that there is an event for which we do have a quantitative upper bound at criticality, namely the two-arms event. Cerf showed that one could deduce from the proof of the uniqueness of the infinite open cluster by Gandolfi, Grimmett and Russo [13] (which was a simplified version of the proof by Aizenman, Kesten and Newman [2]) that there exists a κ = κ(d) such that for all n ∈ N and all p ∈ [0, 1] we have

Pp(two-arms(0, n)) ≤

κ ln n √

n . (1.1)

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boundary of the box Λ(n) = [−n, n]d and their respective open clusters are disjoint in Λ(n).

Cerf improved this bound at criticality by showing there is some explicit γ > 1₂ such that (see also Theorem 3.1)

∃C, ∀n ≥ 1, _P_p_c(two-arms(0, n)) ≤ C

nγ. (1.2)

It must be noted that in two dimensions, on the triangular lattice, it is rigorously known that two-arms probability behaves as n−5/4+o(1), see [26]. For bond percolation on the square lattice in two dimensions, a bound of the order n−α with exponent α slightly bigger than 1 is known. This was implicitly in [21] and proved with different arguments in [4]. In high dimen-sions (d ≥ 19), the two-arms probability behaves as n−4 [22]. We state the result (1.2) for all dimensions as the proof works more or less simultaneous for them all.

For results related to the convergence rates of central limit theorems concerning minimal spanning trees, Chatterjee and Sen [8] wrote that they could use a bound of the form (1.2) if it holds uniformly in p on some open interval containing pc. Now, they use (1.1). The result of Cerf (1.2) can be

shown to hold uniformly in p ≥ pcin a straightforward way.

We extend the result of Cerf in Chapter 3 by showing that it holds uniformly in p. That is, we show in Theorem 3.3 that

∃C : ∀p ∈ [0, 1], ∀n ≥ 1, _P_p(two-arms(0, n)) ≤ C

nγ. (1.3)

The proof we found follows from splitting the interval [0, 1] into [0, pn]∪[pn, 1]

for each n, where pn < pc is chosen carefully. We show that the results by

Cerf can be extended to hold above pn and we use a different argument to

deduce it below pn. We note that the exponent in (1.3) is not affected by

our arguments and is thus the same as the one from Cerf (1.2)

We continue with the two-arms event in Chapter 4, but now turning to a lower bound. We prove in Theorem 4.1 that

∃C > 0 : ∀n ≥ 1, Ppc(two-arms(0, n)) ≥

C nd2_+4d−4.

Our exponent η := 2d2+ 4d − 4 is probably far from optimal, but we have found no polynomial lower bound in the literature (for d ∈ {3, . . . , 6}). A more comprehensive introduction of the problem and its proof can be found at the beginning of Chapter 4.

We finish Chapter 4 by proving, using essentially the same arguments, that there is an exponent η0 < η = 2d2+4d−4 such that if Ppc(two-arms(0, n))

is of the order at most n−η0, then there is no percolation at criticality. We refer to Theorem 4.9 below. It has been rumored1that the late Harry Kesten

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has proved a much stronger result. For d = 3, this is: if the two-arms prob-ability is upper bounded by n−2, then there is no percolation at criticality. Unfortunately, nobody seems to have an idea about the arguments he used.

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Chapter 2

Mathematical framework

and preliminary results

We will provide some basics of percolation theory, together with some nota-tions we will use throughout this thesis. We have decided to omit most of the proofs in this chapter. For a comprehensive introduction into percolation theory, we refer to [14, 15].

2.1 The framework

Let us start by defining the underlying graph used for standard percolation. Fix d ≥ 2 in what follows (the case d = 1 is trivial). We consider Zd_together

with the edge-set

Ed= {{x, y} : x, y ∈ Zd, kx − yk1 = 1}

which in words coincides with nearest neighbor connections. These together build the square lattice Ld = (Zd, Ed). In percolation, vertices are often called sites and edges are called bonds. We will write x ∼ y when x and y are neighbors, i.e. when {x, y} ∈ Ed.

We write [n, m] to mean [n, m] ∩ Z whenever it is clear from the context that we are talking about subsets of the integers. With this, we define for n ∈ N the box Λ(n) = [−n, n]d; an object used very often.

Let A ⊂ Zd. We define the inner boundary of A as

∂A = {x ∈ A : there exists y ∈ Zd\ A with y ∼ x}.

Similarly, we will define the outer boundary as (equivalently) ∂outA = ∂(Ac) or

∂outA = {x ∈ Zd\ A : there exists y ∈ A with y ∼ x}. With this we set the ‘discrete closure’ of A as A = A ∪ ∂outA.

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For site percolation, we consider the set Zd and look at configurations on Ω = {0, 1}Zd_{. For ω ∈ Ω, we say that a site x is open whenever ω}

x = 1

and closed if ωx = 0. One can view ω as a subgraph of Ld, where we have

‘deleted’ all closed sites: Vω = {x ∈ Zd : ωx = 1}. The corresponding edge

set is then the one induced by Vω (i.e. Eω= {{x, y} ∈ Ed: x, y ∈ Vω}).

For Bernoulli (site) percolation with parameter p, we want to construct a probability space that satisfies the following. Each site x ∈ Zd is open with probability p (and hence closed with probability 1 − p) and this is in-dependent of the values of the other sites. To be precise, we will consider the probability space (Ω, F , Pp), where F is the σ-algebra generated by the

events {ωx = 1} for x ∈ Zd and Pp the corresponding product measure

(where each site corresponds with a Bernoulli random variable with param-eter p). A similar definition holds for ‘bond’ percolation, where we replace Ω with {0, 1}Ed.

We will say that x is connected to y whenever there exists v1, . . . , vn∈ Zd

with x = v1, . . . , vn = y, such that {vi, vi+1} ∈ Ed for each i and vi is open

for each i ≥ 2. We write x ↔ y in this case. Note that the event x ↔ y is thus independent of the value of x.We sometimes restrict to paths staying within a certain set C, and write ‘x↔ y’ or ‘x ↔ y in C’ for the event thatC there exists a path x = v1, . . . , vn= y such that {vi, vi+1} ∈ Ed, each vi∈ C

and each vi is open for i ≥ 2. We can also be interested in the connection

between sets, so for A ⊂ Zd we would write x ↔ A for the event that x is connected to some y ∈ A.

The open cluster C(x) of x is defined as follows. If x is closed, C(x) = ∅. If x is open, C(x) = {y ∈ Zd: x ↔ y}.

We will be specifically interested in the events ‘there exists an infinite open cluster’ and ‘x belongs to an infinite open cluster’. In the latter case, we will write x ↔ ∞. Note that this is the same as {|C(x)| ≥ n for each n ∈ N}. This shows why these events are measurable with respect to the σ-algebra F .

Define for j ∈ N≥0 and n, m ∈ N≥0 the ‘j-arm event’ Aj(m, n) as the

event Λ(m) ↔ ∂Λ(n) by j different clusters in Λ(n). To be precise, let C denote the collection of open clusters in Λ(n), then the j-arm event is:

(

∃C1, . . . , Cj ∈ C : ∀i : Ci∩ Λ(m) 6= ∅, Ci∩ ∂Λ(n) 6= ∅

∀i 6= k : Ci∩ Ck= ∅

)

,

In this text, we are particularly interested in the two-arms event A2(m, n).

To emphasize this, we write two-arms(x, n) for the event that two neighbors of x are connected to x + ∂Λ(n) by two disjoint clusters in x + Λ(n). Increasing events, FKG-Harris and the BK-inequality

Intuitively, an event is increasing whenever opening up sites makes the event more likely. To be precise, we call an event A ∈ F increasing whenever

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ω ≤ ω0 and ω ∈ A implies ω0 ∈ A. Here ‘≤’ denotes the component-wise partial order. An event is called decreasing if its complement is increasing. A classical example of an increasing event is 0 ↔ x. With this, we have the famous FKG-Harris inequality (which holds in a more general setting than product measures).

Theorem 2.1 (Fortuin, Kasteleyn, Ginibre [12]; Harris [19]). Suppose that A and B are both increasing (or both decreasing) events, then

Pp(A ∩ B) ≥ Pp(A)Pp(B).

For an increasing event A (depending on a finite number of sites), the next theorem asserts that the map p 7→ Pp(A) is strictly increasing, unless

Pp(A) is trivial for all p. This result will be used later in combination with

the following observation on continuity. If A depends on a finite number of sites, say A depends on {x1, . . . , xn} ⊂ Zd, then by definition of Pp we have

Pp(A) =

X

ω∈A

p|{i:ωxi=1}|_{(1 − p)}|{i:ωxi=0}|_.

This shows that p 7→ Pp(A) is continuous when A depends on a finite number

of sites.

Theorem 2.2. Let A ∈ F be an increasing event, depending on a finite number of sites. Then either for each p, p0 ∈ [0, 1] with p < p0 _{we have that}

Pp(A) < Pp0(A)

or Pp(A) ∈ {0, 1} for each p.

By continuity of p 7→ Pp(A), the last statement of the theorem asserts

that Pp(A) = 1 or Pp(A) = 0 for all p in the latter case.

Proof. The theorem can be proved using a standard coupling argument. A more or less ‘converse’ inequality of ‘FKG-Harris’ is the BK-inequality. This inequality holds for events depending on a finite number of sites, so we assume for simplicity that Ω = {0, 1}Λ(n) and Pp is the corresponding

Bernoulli product measure. We need to introduce the concept of disjoint occurrence. Let A, B ⊂ Ω and fix some ω ∈ Ω. Then in words: A, B occur ‘disjoint’ in ω if we can find two disjoint subsets L, K ⊂ Λ(n) such that the values of ω on L imply A occurs and the values of ω on K imply B occurs. To be more precise, we define for ω ∈ Ω and L ⊂ Λ(n) the ‘cylinder’ events (following loosely the definitions in [15])

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The disjoint occurrence of A and B is then defined as AB = ( ω ∈ Ω : ∃ disjoint L, K ⊂ Λ(n) with C(ω, L) ⊂ A and C(ω, K) ⊂ B ) .

The BK-inequality was first proved for increasing subsets of Ω by Van den Berg and Kesten [6] and later for general sets by Reimer [25].

Theorem 2.3 (BK-Reimer inequality). For A, B ⊂ Ω, we have Pp(AB) ≤ Pp(A)Pp(B).

We will only use it for increasing events. See for instance [15, Theorem 4.17] for a proof of the case for increasing A, B.

The critical value

We return to the critical value. Consider again the lattice Ldtogether with site percolation and the corresponding measure Pp. Define the value

θ(p) = Pp(0 ↔ ∞) = lim

n→∞Pp(0 ↔ ∂Λ(n)).

The second equality follows from a straightforward argument using {0 ↔ ∂Λ(n)} ⊂ {0 ↔ ∂Λ(n − 1)}. We are interested in the question: when is θ(p) > 0. By Theorem 2.2, the map p 7→ θ(p) is non-decreasing in p.

If there is an infinite open cluster, it is still there after we changed the values of a finite number of sites. From the Kolmogorov 0 − 1-law, it thus follows that

Pp(∃ an infinite open cluster) ∈ {0, 1}

and it equals 1 precisely when θ(p) > 0. We will say that there ‘is percola-tion’ whenever θ(p) > 0. Since we also know that p 7→ θ(p) is non-decreasing, we can define the critical parameter for (site) percolation

pc= pc(d) = inf{p : θ(p) > 0}.

A similar parameter exists for bond percolation, and we will write pbond_c to distinguish them. It is a fundamental result that pc is non-trivial (i.e.

pc∈ (0, 1)) for each d. The definition of pcdoes not provide any information

on whether there is an infinite open cluster at criticality. One of the main questions of the field is: is θ(pc) > 0? This is captured in the following

conjecture.

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The conclusion is known in d = 2 due to arguments going back to Harris and Kesten [19, 20]. This is a special case as their combined results show much more, namely pc(2) = 1₂ where one inequality pc ≥ 1₂ follows from

θ(1₂) = 0. The conjecture is also known to be true for d ≥ 11, due to first Hara and Slade (d ≥ 19) [18] and later [11]. It is the general consensus that the arguments used for high dimensions might be used to show it for d ≥ 7, but do not work for d ≤ 6.

One event that will be of interest is 0 ↔ ∂Λ(n). Of course, if Pp(0 ↔

∂Λ(n)) → 0 as n → ∞, there is no percolation at p. It turns out that below criticality (in all dimensions) we have exponential decay. This result is due to Aizenman and Barksy [1] and (at the same time) Menshikov [24]. Recently, Duminil-Copin and Tassion have given a simpler and shorter proof, which is fairly robust [10].

Theorem 2.5. For each 0 ≤ p < pc, there exists a Ψ(p) > 0 such that for

all n ∈ N

Pp(0 ↔ ∂Λ(n)) ≤ e−nΨ(p).

What follows is related to the proof of exponential decay as in Theorem 2.5, see for instance [15, Section 5.1].

Let n ∈ N and define Rn as the number of sites x ∈ ∂Λ(n) such that

0 ↔ x in Λ(n). Since Rn depends on a finite number of sites, p 7→ Ep[Rn]

is a continuous function, which is strictly increasing in p by a coupling argument, see Theorem 2.2. We get the following result, which goes back to Hammersley [17]. The proof here is different and based on [15, Theorem 5.3], as it uses the BK-inequality (which was not known when Hammersley proved the result).

Proposition 2.6. Let p ∈ [0, 1]. If there exists an n ≥ 1 such that Ep[Rn] = X x∈∂Λ(n) Pp 0Λ(n)↔ x< 1

then there exists some c(p) > 0 such that

Pp(0 ↔ ∂Λ(m)) ≤ e−c(p)m

for all m ∈ N.

Proof. Let n be such that Ep[Rn] = η < 1. If 0 ↔ ∂Λ(2n), then there exists

an x ∈ ∂Λ(n) such that 0 ↔ x in Λ(n) and x ↔ x + ∂Λ(n), where the paths use different sites. By the BK-inequality and translation invariance we find that Pp(0 ↔ ∂Λ(2n)) ≤ X x∈∂Λ(n) Pp 0Λ(n)↔ x_Pp(0 ↔ ∂Λ(n)) ≤ ηPp(0 ↔ ∂Λ(n)) (2.1)

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Suppose now that k ∈ N is such that k ≥ n. Write k = mn + r for some m, r ∈ N ∪ {0} and 0 ≤ r < n. Using (2.1) and recursion, we find that

Pp(0 ↔ ∂Λ(k)) ≤ Pp(0 ↔ ∂Λ(mn)) ≤ ηm≤ η

k n−1

since k ≤ n(m + 1). Since η < 1 the result follows.

2.2 Renormalization Arguments

An important tool in percolation theory (especially at criticality) is renor-malization. Let us present why it is useful, and then give an example. In general, a strategy to prove θ(pc) = 0 could be to find an > 0 and a

se-quence of events (An)∞n=1 each depending on a finite number of sites, such

that for each p we have

(∃n : Pp(An) > 1 − ) ⇐⇒ θ(p) > 0. (2.2)

Since each An depends on a finite number of sites, the map p 7→ Pp(An) is

continuous. So if θ(pc) > 0, one can find a p0 < pcfor which Pp0(A_n) > 1 − .

But then θ(p0) > 0, contradicting p0 < pc. It would thus follow that θ(pc) =

0. The problem is of course to find suitable events An. The idea is to search

for An’s that can be used as building blocks to ‘construct’ a new lattice on

which one can show that there is a positive probability of having an infinite open cluster. If the An’s are chosen carefully, one can then deduce that

there exists an infinite open cluster on the original lattice.

We aim to illustrate the difficulty of finding a suitable sequence An’s.

Let Λ(n) = [−n, n]d be the hypercubic box as usual and let {e1, . . . , ed}

denote the standard basis of Rd_{. Let f : N → N satisfy f (n) ≤ n for each n.}

We will consider the event An as

(i) Λ(f (n)) ↔ (2ne1+ Λ(f (n)) using only sites in Λ(n) ∪ (2ne1+ Λ(n)).

(ii) There is at most one cluster in Λ(n) connecting Λ(f (n)) with ∂Λ(n). In other words, the event A2(f (n), n) does not occur.

We just sketch the idea here. Suppose that we know Pp(An) > 1 − for some

n. Then we could deduce from (i), using the FKG-Harris inequality, that Λ(f (n)) is also connected to σ2nei+Λ(f (n)) for each i = 1, . . . , d; σ ∈ {−, +}

with large probability and these connections stay within the box Λ(3n). Due to (ii), it follows that these 2d paths are connected even within the box Λ(n). See Figure 2.1 for some insight. By translation invariance, this holds for each pair of boxes x + Λ(f (n)), x + Λ(n) for x ∈ 2nZd.

We can use this to define a site percolation process on this new lattice. Suppose that x ∈ 2nZd. We say that x is open only if the analog of (i) occurs in all directions σei for i ∈ {1, . . . , d} and σ ∈ {+, −} and the analog

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Figure 2.1: Illustration for dimension d = 2. Drawn in green is the event that Λ(f (n)) is connected to σ2nei+ Λ(f (n)) for each σ ∈ {−, +} and i ∈ {1, 2}

following assumption (i). The dotted green lines connect the different paths within the box Λ(n) following the assumption (ii).

of (ii) occurs. Then Ppc(x open) is large. This newly constructed

renor-malized percolation process on the lattice 2nZd is not independent, but the dependency is rather local and this is no problem, see Theorem 2.9 below. We conclude that if the left hand side of (2.2) holds (for sufficiently small ), then the renormalized process percolates and hence the original site process has an infinite open cluster with positive probability, so the right hand side of (2.2) holds.

The reverse implication of (2.2) is, in this example, the hard part. As-sume θ(p) > 0. If f (n) is relatively large, say f (n) = n/2, then it is possible to show that (i) holds with high probability. However (ii) is then not clear at all (for d ≥ 3). On the other hand, if f (n) is not too big (it can be chosen such that (ii) hold with high probability by uniqueness of the infinite open cluster), then it is not clear how to show (i) holds with high probability.

This illustrates the difficulty of finding suitable An’s. Some other

(pos-sibly) useful examples that are related to the conjecture θ(pc) = 0 can be

found in [9]. Moreover, two famous results that use the renormalization technique (of course, adapted to the respective settings - which is itself not straightforward) are captured in the next two theorems. We do not prove them here.

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Zd−1×Z+where Z+is the set of non-negative integers. For an infinite subset

A ⊂ Zdfor which the induced graph w.r.t. Ldis connected, we define pc(A)

as the critical probability for site percolation. For instance pc= pc(Zd). We

also write

θA(p) = Pp(0 ↔ ∞ on the graph induced by A).

With this, we are ready to state the main results of this section.

Theorem 2.7 (Grimmett and Marstrand [16]). Let d ≥ 2. Then pc =

pc(H).

It must be noted that Grimmett and Marstrand proved a stronger re-sult (which has recently been developed further in a quantitative setting by Duminil-Copin, Kozma and Tassion [9]).

Theorem 2.8 (Barsky, Grimmett and Newman [3]). Suppose that d ≥ 2, then θ_H(pc) = 0.

Actually, the result by Barsky, Grimmett and Newman was proved for pc(H) instead of pc, but due to Theorem 2.7 this is the same.

2.3 Domination of dependent families

We will use later a ‘static’ renormalization scheme, in which we will find a bond percolation process on Ld that is not fully independent, but only ‘locally’ dependent. To be precise, let Y denote a family of random variables on Ed,

Y = (Ye: e ∈ Ed).

We will be interested in the minimal distance between A, B ⊂ Ed w.r.t. to the graph-theoretic distance δ induced by Ld_{. The minimal distance is}

min{δ(x, y) : ∃e ∈ A and f ∈ B such that x ∈ e, y ∈ f }. We call Y a k-dependent family of random variables (or simply k-k-dependent) if for every pair of sets A, B ⊂ Ed_{for which the minimal distance is strictly larger than}

k, the families (Ye : e ∈ A) and (Ye: e ∈ B) are independent.

We will restrict to Bernoulli families Y (that is, each Ye ∈ {0, 1} with

probability one) and say that it stochastically dominates Z = (Ze : e ∈

Ed) if for each increasing (w.r.t. the component wise ordering) function f : {0, 1}Ed → {0, 1}Ed

we have

E[f (Y )] ≥ E[f (Z)].

In what follows, we let Zη = (Zeη : e ∈ Ed) be an i.i.d. family of

Bernoulli random variables with parameter η ∈ [0, 1]. The following result links k-dependent families of Bernoulli random variables with independent families.

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Theorem 2.9 (Liggett, Schonmann and Stacey [23], see also [14], Theorem 7.65). Let d, k ≥ 1. There exists a non-decreasing map π : [0, 1] → [0, 1] satisfying π(η) → 1 as η → 1 such that the following holds. Let Y = (Ye :

e ∈ Ed) be a k-dependent family of Bernoulli random variables, with P(Ye= 1) ≥ η for each e ∈ Ed.

Then Y stochastically dominates Zπ(η) = (Zeπ(η): e ∈ Ed).

It must be noted that the result in [14] is stated for site percolation, but it can be extended in a straightforward manner to bond percolation. The result in [23] holds for more general graphs (with bounded degree), so the adaptation to bond percolation is more straightforward.

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Chapter 3

A uniform upper bound on

the two-arms probability

We focus on results related to the paper by Cerf [7]. He provided a way to improve the upper bound for the two-arms probability (and thus a lower bound on the two-arms exponent). We will also use two intermediate results (Lemma 3.8 and Proposition 3.9 below) in Chapter 4.

In this chapter, we present a generalization of the results by Cerf, mainly adding a uniform bound in p, whereas the bound of Cerf was stated for p = pc. To be precise, the following theorem can be obtained in a straightforward

manner from the proof of Theorem 1.1 in [7].

Theorem 3.1 (See Theorem 1.1 in [7]). Let d ≥ 2 and define γ∞= γ∞(d) = 2d2_+3d−3

4d2_+5d−5. For each γ < γ∞, there exists a c(d, γ) > 0 such that for all n ∈ N

we have

Ppc(two-arms(0, n)) ≤ c(d, γ)

1 nγ.

Note first that γ∞(d) > 1₂ for each d ≥ 2. Also note that in Theorem 3.1,

the map p 7→ Pp(two-arms(0, n)) is not necessarily bounded by c(d, γ)n−γ

uniformly in p (although it follows immediately from the arguments of Cerf that the bound holds uniformly on p ≥ pc). Chatterjee and Sen [8]

specifi-cally ask for a bound of the form Pp(two-arms(0, n)) ≤ c(d, γ)n−γ, holding

for all n and uniformly in p on some interval containing pc, with γ > 1₂. They

could use this result to further improve convergence rates on central limit theorems for the length of minimal spanning trees, see also the discussion in the remarks following Lemma 5.2 in [8].

As noted by Cerf [7] and used by Chatterjee and Sen [8], one could distill from the proof of the uniqueness of the infinite cluster by Gandolfi, Grimmett and Russo [13] (which was a simplified version of [2]) that there is some κ = κ(d) such that for all p ∈ [0, 1] and all n ∈ N we have

κ ln n √

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Thus, getting the bound in Theorem 3.1 to hold uniformly in p would improve this result. A result of this kind, that was proved (using Cerf’s results) in a recent preprint by Duminil-Copin, Kozma and Tassion1, is the following. Recall from Chapter 2 that A2(m, n) denotes the event that there

are two disjoint clusters in Λ(n), connecting Λ(m) with ∂Λ(n).

Proposition 3.2 (Proposition 1 in [9]). Let d ≥ 2. There exists an α = α(d) ∈ (0, 1) and a constant c = c(d) > 0 such that for all n ∈ N and p ∈ [0, 1] we have

Pp(A2(nα, n)) ≤

c nα.

Although Proposition 3.2 as sated in the current version of [9] does not provide an explicit α = α(d), the arguments they sent us do so. These arguments could also be used to get a uniform version of Theorem 3.1 (in p), but with a weaker exponent ‘γ∞’.

We prove uniformity in p of Theorem 3.1 in such a way that we obtain the same exponent γ∞, see Theorem 3.3 below. We note that our result requires

considerably more work than the proof of Proposition 3.2 by Duminil-Copin, Kozma and Tassion, as we need to update the intermediate results by Cerf [7]. These updates are rather subtle and are presented in full detail below. Theorem 3.3 (Uniform version of Theorem 1.1 in [7]). Let d ≥ 2 and set γ∞ = 2d

2_+3d−3

4d2_+5d−5. For each γ < γ∞, there exists a c(d, γ) > 0 such that for

all n ∈ N and all p ∈ [0, 1] we have

Pp(two-arms(0, n)) ≤ c(d, γ)

1 nγ.

To prove this theorem, we update the two-point lemma (see Lemma 3.8) and then show that the other results still work. To that end, we split the interval [0, 1] into two parts [0, pn) and [pn, 1], where pn is a special p < pc

depending on n (defined in Section 3.2) such that the one-arm probability Pp(0 ↔ Λ(n)) decays to zero in a fast and ‘controlled’ way for p < pn and

such that for p ∈ [pn, 1] the remaining computations of Cerf are still valid.

3.1 The starting point: an inequality.

In what follows, we need a few additional definitions. Let C = C(n, l) be the collection of open clusters in Λ(n + l) that connect Λ(n) with ∂Λ(n + l). To be precise

C =C open cluster in Λ(n + l) : C ∩ Λ(n) 6= ∅, C ∩ ∂Λ(n + l) 6= ∅ . (3.1) We are now ready to state what Cerf calls the ‘central inequality’.

1_{The precise argument is omitted in the current Arxiv version, but they showed it in}

private communication after we presented our argument to ask if they had the same ideas. For the result they need, their argument is simpler.

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Lemma 3.4 (Lemma 5.2 in [7]). For all p ∈ (0, 1) and all n ∈ N, l ∈ N∪{0} we have that Pp(two-arms(0, 2n + l)) (3.2) ≤ 2d ln n p|Λ(n)|Epp|C| + 4d p(1 − p)|Λ(n)| 2_{exp − 2(ln n)}2_p2_{(1 − p)}2

The proof of this lemma essentially follows the structure of the proof of the uniqueness of the infinite cluster by Gandolfi, Grimmett and Russo [13] (which was a simplified version of the proof by Aizenman, Kesten and Newman [2]). We sketch shortly the difference in the arguments. For a detailed proof of Lemma 3.4, we refer to [7, Sections 3-5].

Gandolfi, Grimmett and Russo showed that the probability that a site belongs to the outer boundary of two different infinite open clusters equals zero (which is related to a two-arm event). They did so in the following way. Define ˜C as the set of open clusters in Λ(n) that contain at least one point in the boundary ∂Λ(n). Define the (random) set ˜H that contains all points in Λ(n) that are in the outer boundary of two different clusters C1, C2∈ ˜C,

i.e. ˜ H = [ C1,C2∈ ˜C C16=C2 ∂outC1∩ ∂outC2.

They then provide a quantitative upper bound of expected number of sites Ep| ˜H| (as a function of n) and use this to show that

lim sup

n→∞

1

|Λ(n)|Ep| ˜H| ≤ 0.

Cerf changes these arguments by introducing a parameter l and annulus crossings. Instead of ˜C, he defines the collection C = C(n, l) as above in (3.1) and H as the set of points in Λ(n) that are in the outer boundary of two different sets C1, C2 ∈ C, so

H = [

C1,C2∈C

C16=C2

∂outC1∩ ∂outC2∩ Λ(n).

He deduces the central inequality (Lemma 3.4) from this, using similar ar-guments as Gandolfi, Grimmett and Russo. The new parameter l and the corresponding annulus crossings allow Cerf (as we will see below in Section 3.5) to then bound the expectation Epp|C| in (3.2) in terms of two-arms

events and then apply (3.2) to these two-arms events. This ‘bootstrap’ idea eventually led to Theorem 3.1.

Taking l = 0 and using that |C| ≤ |∂Λ(n)|, the central inequality offers the following lemma. It was pointed out by Cerf [7] that this could also be distilled more or less directly from [13].

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Lemma 3.5 (Proposition 5.3 in [7]). Let d ≥ 2 and δ > 0. There exists a constant κ = κ(δ, d) such that for all p ∈ [δ, 1 − δ] and all n ≥ 1 we have

κ ln n √

n .

Note that this bound is uniform in p on some interval containing pc.

3.2 Definition of p

n

and an immediate result.

We want to find a value pn∈ [0, pc) such that on the one hand we can use the

result by Cerf and on the other hand have explicit control on the one-arm decay below the value. It will, moreover, turn out to be convenient to define the pn’s such that they are non-decreasing in n.

Recall from Section 2.1 that Rn denotes the number of sites x ∈ ∂Λ(n)

such that 0 ↔ x in Λ(n). Fix n ∈ N. Note that E0[Rn] = 0 . By Theorem

2.5 and Proposition 2.6 we have that Epc[Rn] ≥ 1. Because p 7→ Ep[Rn] is

continuous and strictly increasing, the following is well defined.

Definition 3.6. Let n ∈ N. We define pn as the smallest p ∈ (0, pc) such

that ∀k ≤ n, _E_p[Rk] = X x∈∂Λ(k) Pp 0Λ(k)↔ x≥ 1 e. (3.3)

We will need the next, immediate result of this definition at the end of the proof of Theorem 3.3.

Lemma 3.7. Let ξ ∈ (0, 1). Fix n ∈ N and p ∈ [0, pnξ). We have

Pp(0 ↔ ∂Λ(n)) ≤ e−n

1−ξ₊₁

.

It will be crucial that there is no constant depending on p in the right-hand side.

Proof. The proof is essentially the same as the proof of Proposition 2.6. We highlight the differences.

Fix ξ ∈ (0, 1) and n ∈ N. Since 0 ↔ ∂Λ(n) is increasing, we get from Theorem 2.2 that for all p ∈ [0, p_nξ)

Pp(0 ↔ ∂Λ(n)) ≤ Pp_nξ(0 ↔ ∂Λ(n)).

We can thus restrict to showing the desired bound in p_nξ. By definition of

p_nξ, there exists a k ≤ nξ such that

Ep_nξ[Rk] =

1 e.

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Indeed, if this would not exist, we could find a smaller p ∈ (0, pc) satisfying

relation (3.3). Fix this k and set η = 1_e. By arguments completely similar to those in the proof of Proposition 2.6, we obtain that

Pp_nξ(0 ↔ ∂Λ(n)) ≤ ηn

1−ξ₋₁

= e−n1−ξ+1.

3.3 A uniform bound for the two-point function.

In this section, we restrict to the case p ≥ pn. The following lemma is an

extension of Lemma 6.1 in [7], where the result is stated to hold for p = pc

(and thus by monotonicity also for p > pc).

Lemma 3.8 (Uniform version of Lemma 6.1 in [7]). Let d ≥ 2. There exists a positive constant C = C(d) such that for all n ≥ 1 and p ∈ [pn, 1] we have

Pp(x ↔ y in Λ(2n)) ≥

C n2(d−1)d

for all x, y ∈ Λ(n).

Proof. Let d ≥ 2, n ∈ N and p ∈ [pn, 1]. By definition of pn and the fact

that p 7→ Ep[Rn] is non-decreasing, we obtain that for all 1 ≤ k ≤ n we have

X

x∈∂Λ(k)

Pp(0 ↔ x in Λ(k)) = Ep[Rk] ≥

1 e.

In particular, for each k, there exists a point x∗_k∈ ∂Λ(k) such that Pp(0 ↔ x∗k in Λ(k)) ≥

1 e|∂Λ(k)| ≥

c(d) kd−1,

for some constant c(d) > 0. By rotation invariance, we may assume that x∗_k∈ {x ∈ Λ(k) : x₁ = k}.

Let {e1, . . . , ed} denote the standard basis for Rd and let x, y ∈ Λ(n).

Assume for now that |xi− yi| is even for each i = 1, . . . , d and define 2z =

x − y. Then |zi| ≤ n and zi ∈ Z. Define (v)0 = 0 ∈ Zd and set (v)i =

x −Pi

j=12ziei, so that (v)i = (y1, . . . , yi, xi+1, . . . , xd).

The idea is to travel from (v)i to (v)i+1 for each i. We then exploit

translation and rotation invariance of our model, in combination with the special x∗_k’s used as follows. Clearly, if 0 ↔ x∗_k and x∗_k ↔ 2ke1, then 0 ↔

2ke1. This will give the desired result.

By translation invariance and FKG-Harris inequality we have Pp(x ↔ y in Λ(2n)) ≥ Pp \d i=1 {(v)_i−1↔ (v)_i in Λ(2n)} ≥ d Y i=1 Pp 0 ↔ 2ziei in Λ(n) ∪ (2ziei+ Λ(n)).

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Now we will use our special x∗_k’s for 1 ≤ k ≤ n. Note that Pp 0 ↔ 2ziei in Λ(n) ∪ (2ziei+ Λ(n)) = Pp 0 ↔ 2|zi|e1 in Λ(n) ∪ (2|zi|e1+ Λ(n)) ≥ Pp(0 ↔ x∗|zi|in Λ(|zi|)Pp(x ∗ |zi|↔ 2|zi|e1 in 2|zi|e1+ Λ(|zi|)) ≥ c(d) |z_i|d−1 2 ,

where the first equality follows from rotation invariance and the first in-equality from FKG-Harris. We thus get that

Pp(x ↔ y in Λ(2n)) ≥ d Y i=1 c(d) |zi|d−1 2 ≥ c(d) 2d n2d(d−1)

as required. The only additional assumption we made was that each |xi− yi|

was even. To overcome this problem, we note that one can always walk a few steps from y in the direction of x to find a ˜y for which xi− ˜yi is even (or

0, in which case the result is trivial). Of course, this ˜y can be picked close to y. To be precise, we can find ˜y ∈ Λ(n) such that for each i = 1, . . . , d we have

|xi− ˜yi| is even |yi− ˜yi| ≤ 1.

In this case, the probability of the event y ↔ ˜y in Λ(2n) is trivially lower bounded by pd ≥ (p1)d > 0 as n 7→ pn is non-decreasing. If we thus set

C = C(d) = c(d)2d(p1)d, we get that

C n2d(d−1).

Remark 3.1. The original version of Lemma 3.8 is: there is some C = C(d) > 0 such that for all n ∈ N we have

Ppc(x ↔ y in Λ(2n)) ≥

C n2d(d−1)

for all x, y ∈ Λ(n), see [7, Lemma 6.1]. This has recently been sharpened by Van den Berg and Don (preprint) [5], where the right-hand side is of the order n−d2. We note that it seems possible to modify the arguments above to extend the result of Van den Berg and Don to: there exists a C = C(d) > 0 such that for all n ∈ N and p ∈ [˜pn, 1] we have that

C nd2

for all x, y ∈ Λ(n). It will follow below that the 9n instead of the 2n is of no importance. The definition of ˜pn would then differ (a bit) from the

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We also note that, although the updated result in [5] is interesting on itself and helpful for the results in Chapter 4, it only improves the exponent in Theorem 3.3 a little due to the geometric convergence as we will see in Lemma 3.11.

3.4 A relation between two ‘two-arm’ events.

In this section, we will sketch the proof of an important result (Proposition 3.9 below) that will be used in the subsequent chapter as well. Note that the result presented here was already in [7] and we do not need a different version. For this reason, we omit a formal proof, but instead offer a sketch. For all the details, we refer to [7, Section 7].

Recall from Section 2.1 that A2(m, n) is the event that there are two

disjoint (open) clusters connecting Λ(m) with ∂Λ(n).

Proposition 3.9 (Corollary 7.3 in [7]). Let n ≥ 1 and l ≥ n. For each p ∈ (0, 1) we have that Pp(A2(n, n + l)) ≤ 39d p n4d−2Pp(two-arms(0, l − n)) inf{Pp(a ↔ b in Λ(2n)) : a, b ∈ ∂Λ(n)}

Sketch of the proof. Fix n ∈ N and l ≥ n. Suppose that the event A2(n, n + l) occurs, then there are two points a, b ∈ ∂Λ(n) such that the

respective clusters C(a), C(b) are disjoint in Λ(n + l) and both clusters in-tersect the boundary ∂Λ(n + l). Fix these a, b.

There are essentially two possibilities. Either the two clusters are ‘close’ to each other, or not. Suppose they are close: ∂outC(a) ∩ ∂outC(b) ∩ Λ(2n) 6= ∅. In this case, there exists a point z in the latter intersection for which the event two-arms(z, l − n) occurs. See Figure 3.1. Without much work, this would offer the final result (even throwing a lot away).

However, we have no clue how close C(a), C(b) need to be. Hence, we will need to do a little more work in the case they are not ‘close’. Suppose

∂outC(a) ∩ ∂outC(b) ∩ Λ(2n) = ∅. (3.4) We want to know how likely the outer boundaries of C(a), C(b) are ‘con-nected’. Suppose to that end that A, B ⊂ Zd are such that C(a) = A and C(b) = B.

Let D, E be two subset of Zdsuch that D ∩ E = ∅. In this case, ∂outD ↔ ∂outE is independent of the value of the sites in D ∪ E. If, moreover, a ∈ D and b ∈ E we clearly have

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Figure 3.1: The clusters C(a) and C(b) are good, so their outer boundaries touch within the box Λ(n + k). For the point z, the event two-arms(z, l − k) occurs.

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Figure 3.2: The clusters C(a) and C(b) are not close, and a path connecting ∂outC(a) and ∂outC(b) is drawn. Opening up the neighbor zb implies the

event two-arms(za, l − n).

The last observation we need to make is that the event C(a) = A depends only on the value of the sites in A, so we get that

Pp(a ↔ b in Λ(2n))Pp(C(a) = A, C(b) = B)

≤ Pp(C(a) = A; C(b) = B, ∂outC(a) ↔ ∂outC(b) in Λ(2n))

We thus see that the cost of C(a) being ‘almost connected’ to C(b) is Pp(a ↔ b in Λ(2n)), see Figure 3.2. To be precise, there exist elements ua, ub

in ∂out_{C(a) ∩ Λ(2n) and ∂}out_{C(b) ∩ Λ(2n) respectively such that u}

a↔ ub in

Λ(2n). As is visible in the Figure 3.2, opening up the ‘right’ neighbor of ub

that is in ∂outC(b) will imply the event two-arms(za, l − n) from a neighbor

za∈ ∂outC(a) of ua.

Proposition 3.9 now follows from applying the union bound, as we need to sum over all possible a, b ∈ ∂Λ(n) and then over all pairs ua, ub ∈ Λ(2n).

3.5 From the central inequality to an iterative bound.

We go back to the ‘central inequality’, see (3.2) where we will now try to control the expected value Epp|C|. Recall that C denotes the set of clusters

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connecting Λ(n) and ∂Λ(n + l). With Proposition 3.9 and the two-point Lemma 3.8, we will be able to bound Epp|C| in terms of Pp(two-arms(0, n −

k)) for some k. We state the result of this section in the following lemma. Lemma 3.10 (Uniform version of Lemma 9.1 in [7]). Let d ≥ 2 and δ > 0. There exists a positive constant c = c(d, δ) such that for all n, k ≥ 1, with 1 ≤ k ≤ n and p ∈ [pk, 1] ∩ [δ, 1 − δ] we have that

Pp(two-arms(0, 3n)) ≤ c ln n √ n 1 kd−1 + k 2d2_+2d−2 Pp(two-arms(0, n − k)) 1 2 (3.5) The arguments below are the same as those by Cerf [7, Section 8], apart from those after equation (3.7).

Proof. Fix for now the integers k, n, l such that k ≤ n ≤ l. Assume for now that p ∈ [δ, 1 − δ]. We will cover the boundary of the box Λ(n) (from the inside) with boxes of size k. Let (Λi : i ∈ I) be a collection of boxes that are

translates of Λ(k) so that ∂Λ(n) ⊂ S

i∈IΛi, for some index set I. We will

not care about possible intersections of such boxes. We will need at most |I| ≤ 2d2n k d−1 = d2dn k d−1

such boxes, as Λ(n) has 2d faces. Fix such a covering indexed by I of the boundary ∂Λ(n). Now a box Λi = yi+ Λ(k) is called bad if there exist (at

least) two different clusters within the box yi+ Λ(k + l) that connect Λi with

yi+ ∂Λ(k + l). In this case, a translate of the event A2(k, k + l) occurs. The

box Λi is called good otherwise.

It follows that Ep {i : i ∈ I, Λ_i is bad}= E_p h X i∈I 1Λiis bad i = |I|Pp(A2(k, l + k)). (3.6)

Now note that any C ∈ C intersects the boundary of Λ(n), hence it must intersect some box Λi in the collection (Λi : i ∈ I). If a box is good, it can

only intersect one element of C. If a box is bad, it can intersect at most |∂Λ(k)| elements of C. In particular

|C| ≤ |{i : i ∈ I, Λi is good}| + |∂Λ(k)| · |{i : i ∈ I, Λi is bad}|.

Taking expectation and using the relation in (3.6) we get Ep|C| ≤ |I| + |∂Λ(k)| · |I| · Pp(A2(k, l + k)) ≤ d2dn k d−1 + d2dn k d−1 |∂Λ(k)|Pp(A2(k, k + l)) ≤ ˜cn k d−1 + ˜cnd−1Pp(A2(k, k + l)),

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for some ˜c depending only on d. By Proposition 3.9 to control the second term, we obtain Ep|C| ≤ ˜c n k d−1 + ˜cnd−13 9d p k4d−2Pp(two-arms(0, l − k) inf{Pp(a ↔ b in Λ(2k)) : a, b ∈ ∂Λ(k)}

Since p ≥ δ, we can set c = c(d, δ) = ˜c39d_δ to obtain that

Ep|C| ≤ c n k d−1 + cn d−1_k4d−2 Pp(two-arms(0, l − k) inf{Pp(a ↔ b in Λ(2k)) : a, b ∈ ∂Λ(k)} .

By Jensen’s inequality, we also have Epp|C| ≤ (Ep|C|)

1

2. We thus obtain

from the central inequality (3.2) that Pp(two-arms(0, 2n + l)) ≤ 2d ln n p|Λ(n)| c n k d−1 + cn d−1_k4d−2 Pp(two-arms(0, l − k) inf{Pp(a ↔ b in Λ(2k)) : a, b ∈ ∂Λ(k)} !1 2 + 4d p(1 − p)|Λ(n) 2_{| exp − 2(ln n)}2_p2_{(1 − p)}2_.

Pick l = n. We can now update c = c(d, δ) (still not depending on p, because both p and 1 − p stay above δ by assumption) to get

Pp(two-arms(0, 3n)) ≤ cln n√ n 1 kd−1 + k4d−2Pp(two-arms(0, n − k) inf{Pp(a ↔ b in Λ(2k)) : a, b ∈ ∂Λ(k)} !1₂ . (3.7)

Up to this point, all inequalities were valid for all p ∈ [δ, 1 − δ]. We will invoke Lemma 3.8. It follows that for all p ∈ [pk, 1] ∩ [δ, 1 − δ] we have that,

for some yet updated c (which only depends on d, δ)

Pp(two-arms(0, 3n)) ≤ c ln n √ n 1 kd−1 + k 2d2+2d−2 Pp(two-arms(0, n − k)) !1₂ ,

which is the desired inequality.

3.6 Iteration and the final bound.

With the last presented lemma, we are in the position to iterate what we have done so far and are almost ready to join the p above and below pn.

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Lemma 3.11. Let d ≥ 2 and δ > 0. Set γ∞= 2d

2_+3d−3

4d2_+5d−5 and ξ = _2d2_+3d−31 .

For each γ < γ∞, there exists a c = c(d, δ, γ) such that for each n ∈ N,

p ∈ [p_nξ, 1] ∩ [δ, 1 − δ] we have that

c nγ.

Proof. We follow again Cerf’s arguments, see [7, Section 9] and adapt them essentially to the case p ∈ [p_nξ, p_c].

Let d ≥ 2. Suppose that we know that there exists some c0 = c0(d, δ) such that for all n, we have that for all p ∈ [p_nξ, 1] ∩ [δ, 1 − δ]

Pp(two-arms(0, n)) ≤ c0

ln(n)β nγ ,

for some γ, β > 0 with γ < 1. We will use (3.5). Pick k = nν _{in (3.5), where}

ν is defined by

ν = γ

2d2_{+ 3d − 3} ≤ ξ.

Since k 7→ pk is increasing by definition of pk, we thus get that pnν ≤ p_nξ.

Using also Lemma 3.10 we find that for all n ∈ N, p ∈ [pnξ, 1] ∩ [δ, 1 − δ] we

have Pp(two-arms(0, 3n)) ≤ c ln n √ n 1 nν(d−1) + c0(ln(n − nν))βnν(2d2+2d−2) (n − nν₎γ !1₂

Hence, there exists some updated c00(d, δ) for which

Pp(two-arms(0, 3n)) ≤ c00ln n √ n (ln n)β nν(d−1) 1₂ . If we thus set γ0 = 1 2 + 1 2ν(d − 1) = 1 2 + d − 1 4d2_{+ 6d − 6}γ,

and also use monotonicity in n of n 7→ Pp(two-arms(0, n)) we find for a yet

again slightly modified c00= c00(d, δ) that

c00(ln n)β+1

nγ0 ,

which hold for all n and all p ∈ [p_nξ, 1] ∩ [δ, 1 − δ]. Therefore, we can keep

updating γ with γ0 iteratively, if we have a given starting point, since also the function n 7→ pn is non-decreasing by definition. This starting point is

in Lemma 3.5, stating

κ ln n √

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where κ = κ(d, δ) and does not depend on p. This bound holds for all n ≥ 1 and all p ∈ [δ, 1 − δ], so in particular it holds for all n ∈ N and p ∈ [p_nξ, 1] ∩ [δ, 1 − δ].

Now we can define a sequence (γi)∞i=0inductively, setting γ0= 1₂ and

γi =

1 2 +

d − 1

4d2_{+ 6d − 6}γi−1.

Using the iterative argument of above, noting that all γi’s are below 1, we

can find a sequence of constants αi = αi(d, δ) such that for each n ≥ 1,

p ∈ [p_nξ, 1] ∩ [δ, 1 − δ] we have that

αi(ln n)i+1

nγi . (3.8)

The sequence γi is increasing by definition and converges geometrically to

γ∞, where γ∞ is as in the statement of Lemma 3.11.

Let γ < γ∞, then there exists an i such that γi > γ. We thus get, using

also (3.8), the existence of a c(d, γ, δ) such that

αi(ln n)i+1

nγi ≤

c(d, γ, δ) nγ

holds for all p ∈ [p_nξ, 1] ∩ [δ, 1 − δ] and all n ∈ N. This is the desired

result.

We are now ready to show the final result.

Proof of Theorem 3.3. Fix δ > 0, and let γ < γ∞, with γ∞ as in Lemma

3.11 and set c = c(d, δ, γ), ξ as in Lemma 3.11. We thus have for each n ∈ N, p ∈ [p_nξ, 1] ∩ [δ, 1 − δ] that

c nγ.

We turn to the case p < p_nξ. By Lemma 3.7 and using a trivial upper bound,

we get

Pp(two-arms(0, n)) ≤ Pp(0 ↔ ∂Λ(n)) ≤ e−n

1−ξ₊₁

≤ c nγ

whenever n is large enough. So we may define c0 = c0(d, δ, γ) such that for all n ∈ N and all p ∈ [0, pnξ) ∪ [p_nξ, 1 − δ] we have

c0 nγ.

We are left to handle the case p ≥ 1 − δ. We can do so based on a standard argument. See for instance [15, Theorem 3.2].

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Note that for two-arms(0, n) to occur, there must (at the very least) be a closed path connecting 0 to ∂Λ(n), albeit it is on the slightly more connected ‘star lattice’ (where ‘diagonal’ paths are allowed as well as the nearest neighbor ones). We will write 0 ↔ ∂Λ(n) for this event. Thec vertices set of this ‘star lattice’ is Zd, but now x can be connected to any element in the set x + Λ(1), besides itself. In particular, the degree of this graph is 3d− 1 =: g.

The existence of a closed path 0↔ ∂Λ(n) implies the existence of closedc self-avoiding path of length n. The result then follows by bounding the number of self-avoiding paths on this star lattice of length n in the most crude way: we bound it by g(g − 1)n−1. Now the probability that such a non-self intersection path is closed is (1 − p)n. Hence, if we pick δ small enough, we get the existence of a c = c(d, γ) (by using the fixed δ and plugging it into the previously found c0 = c0(d, γ, δ)) such that

c nγ

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Chapter 4

A polynomial lower bound

for the two-arms probability

We will prove that the two-arms probability is lower bounded by a polyno-mial in n at criticality for all dimensions d ≥ 2. This behavior is different from the regimes p < pc and p > pc. Indeed, if p < pc then the exponential

decay in n of the one-arm event 0 ↔ ∂Λ(n) (see Theorem 2.5) implies of course exponential decay for the two-arms probability. Above criticality, the proof of exponential decay for the two-arms probability is quite different (and non-trivial) but can be found in Grimmett [14, Lemma 7.89].

The result presented here is probably far from optimal. On the trian-gular lattice in two dimensions, it is rigorously known that the two-arms probability is of the order n−54+o(1) [26]. In high dimensions (d ≥ 19), the

order is also known and it equals n−4 [22].

A minor adaptation of the proof of theorem 4.1 shows that if the two-arms exponent is larger than 2d2+2d−2, there is no percolation at criticality. We show this in Section 4.3.

Theorem 4.1. Let d ≥ 2 and η = 2d2 + 4d − 4. There exists a constant C(d) > 0 such that for all n ∈ N we have

Ppc(two-arms(0, n)) ≥

C(d) nη

Corollary 4.2. Let d ≥ 2 and set γ∞ = 2d

2_+3d−3

4d2_+5d−5 and η = 2d2+ 4d − 4.

For each γ < γ∞, there exist C(d, γ) and C(d) > 0 such that for all n ∈ N

we have

C(d)n−η ≤ Ppc(two-arms(0, n)) ≤ C(d, γ)n

−γ_.

Let us give an idea of the proof and its methods. Besides the (hy-per)cubes Λ(n), we will also work with the boxes Bi(m, n) that are of

dif-ferent size in one of the d directions. To be precise, we define Bi(m, n) =

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Denote ˜Fi,1 = {x ∈ Bi(m, n) : xi = 0} and ˜Fi,2 = {x ∈ Bi(m, n) : xi = m}.

See also Figure 4.1 below. We will also define the crossing in the ‘easy’ direction by LRi(m, n) = { ˜Fi,1(m, n) ↔ ˜Fi,2(m, n) in Bi(m, n)}, for each

i ∈ {1, . . . , d}.

The structure of this proof is as follows and requires carefully handling of logical steps. Set ν = 2d2+ 2d − 1. We define the statement (P) to be

∀ > 0, ∀N ∈ N : ∃n ≥ N :

Ppc(LR1(10n, 30n)) ≥ 1 − and Ppc(two-arms(0, n)) ≤ n

−ν_. (P)

The negation of (P) is then

∃ > 0 : ∃N ∈ N : ∀n ≥ N :

Ppc(LR1(10n, 30n)) < 1 − or Ppc(two-arms(0, n)) > n

−ν

. We will show in Section 4.1, using renormalization arguments, that

(P) =⇒ θ_H(pc) > 0, (4.1)

by ‘constructing’ an open cluster in the half-space. Using Theorem 2.8, the famous result by Barksy, Grimmett and Newman [3] stating θ_H(pc) = 0, we

conclude that ‘not (P)’ occurs. On the other hand, we will prove ‘directly’ in Section 4.2 that

not (P) =⇒ ∃C(d) > 0 : ∀n ∈ N : Ppc(two-arms(0, n)) ≥ C(d)n

−η_,

(4.2) completing the proof of Theorem 4.1. In the next two sections, we carry out the details.

Remark 4.1. Due to the recent result by Van den Berg and Don [5] (preprint, see also Remark 3.1), the exponent η in Theorem 4.1 can be updated in a straightforward manner to η = d2+ 6d − 3.

4.1 Creating an infinite cluster in the half-space.

In this section, we prove statement (4.1). Based loosely on arguments by Grimmett [14, Section 7.4] and renormalization ideas presented in [9], we will renormalize the lattice Zdto create the infinite cluster in the half-space. Define the (random) set of vertices

Kn= {x ∈ Λ(5n) : x is open, x ↔ ∂Λ(15n)}

and the crossing cluster In = {x ∈ Λ(15n) : x is open, x ↔ Kn in Λ(15n)}.

Define also for each i ∈ {1, . . . , d}, σ ∈ {−, +} and m ∈ N the faces Fi,σ(m) = {x ∈ Λ(m) : xi = σm} (these are the faces of the (hyper)cube

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Definition 4.3. We call the box Λ(15n) good if In is connected in Λ(15n)

and if In∩ Fi,σ(15n) 6= ∅ for each i ∈ {1, . . . , d} and σ ∈ {−, +}.

Consider then the renormalized lattice 6nZd. Let x, y be two neighboring points in 6nZd, so there exists a (unique) 1 ≤ i ≤ d such that |yi− xi| =

6n. Fix this i and assume without loss of generality that yi > xi. Define

Λ0(x, y) = x − 24nei+ Λ(39n). Note that this definition depends on the

precise direction in which x, y differ and on which one is ‘bigger’ in that direction.

Definition 4.4. Let x, y ∈ 6nZd be neighbors. We say that x and y are connected and write x ↔ y if the boxes Λn _x := x + Λ(15n) and Λy :=

y+Λ(15n) are good and the respective crossing clusters are connected within the box Λ0(x, y).

We will show that two neighbors x, y ∈ 6nZd are connected with high probability if we assume (P). This will, for some k not depending on n, define a k-dependent bond percolation on the (renormalized) lattice with arbitrary large connection parameter. Invoking the domination result of Liggett, Schonmann and Stacey [23] (see Theorem 2.9) we get that the renormalized (bond) percolation process dominates an independent (bond) percolation process with parameter strictly above pbond

c = pbondc (H). It thus

follows that the original process must have an infinite cluster in the half-space, giving the desired contradiction. We make this all precise below. Connection on the renormalized lattice.

This subsection is devoted to proving that the probability that two neighbors x, y ∈ 6nZd are connected (as in definition 4.4) can be chosen arbitrarily close to 1.

Lemma 4.5. If (P) is true, then for each > 0, there exists an n ∈ N such that Ppc(x

n

↔ y) ≥ 1 − . Proof. Let > 0. Fix

δ = 1 − (1 −1₆)2d1 .

The reason for this choice of δ will become clear later. We note that δ ∈ (0, 1) whenever is small enough. Recall from Proposition 3.9 and Lemma 3.8 the relation

Ppc(A2(n, n + l)) ≤ c(d)n

2d2+2d−2

Ppc(two-arms(0, l − n)), (4.3)

for some constant c(d). By monotonicity, it is clear that Ppc(two-arms(0, ˜cn)) ≤ Ppc(two-arms(0, n))

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Figure 4.1: The box B1(10n, 30n) in dimension 3 where the gray area is the

face ˜F1,1(10n, 30n) and it is split up in equal parts Cj’s (the enumeration is

arbitrary, we picked one for simplicity).

for each ˜c ≥ 1. We can fix N so large that c(d)52d2+2d−21 n ≤ c(d)15 2d2_+2d−21 n ≤ 1 6,

for all n ≥ N . Set M = 3d−1. It follows from the assumption (P) and the relation (4.3) that there exists some n ≥ N for which

Ppc(A2(15n, 39n)) ≤ 1 6 and Ppc(A2(5n, 15n)) ≤ 1 6 (4.4) and Ppc(LR1(10n, 30n)) ≥ 1 − (δ 2d₎M Fix such an n.

We now split the face ˜F1,1(10n, 30n) into parts of equal size. Let C1 =

{0}×[0, 10n]d−1_{and write C}

2, . . . , CM for the other parts that are translates

of C1, that are mutually disjoint and that satisfy M

[

i=1

Ci= ˜F1,1(10n, 30n).

4.2 Proof of statement (4.2)

We now prove the implication (4.2). This is the simpler part that only requires gluing together three events that happen with large enough proba-bility. We start with a preliminary result that follows from Proposition 2.6 and Theorem 2.5.

Recall the ‘upper’ half-space H = Zd−1× Z+, where Z+ denotes the set

of non-negative integers as usual. We define also the ‘shifted’ half spaces H+(k) = Zd−1 × [k, ∞) and H−(k) = Zd−1 × (−∞, −k]. In particular, H = H+(0).

Lemma 4.7. Let d ≥ 2. There exists a constant c(d) > 0 such that the following holds. Let σ ∈ {−, +}. For all k, m ∈ Z+ such that k < m and all

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x ∈ ∂Hσ(k) we have that Ppc(x ↔ H

σ

(m) in Hσ(k)) ≥ c(d) (m − k)d−1.

Proof. Let d ≥ 2 and k, m ∈ Z+such that k < m. By translation invariance

and symmetry we can assume w.l.o.g. that k = 0, m ≥ 1 and σ = +. By Theorem 2.5 and Proposition 2.6, we get that Epc[Rm] ≥ 1. Equivalently,

we get

X

x∈∂Λ(m)

Ppc(0 ↔ x in Λ(m)) ≥ 1.

Using this last inequality and symmetry, there must exist an x∗_m ∈ F_d,−(m) = [−m, m]d−1× {−m} such that Ppc(0 ↔ x ∗ m in Λ(m)) ≥ 1 |∂Λ(m)| ≥ c(d) md−1, (4.9)

where c(d) > 0 is a constant that does not depend on m. Let {e1, . . . , ed}

be the standard basis for Rd. Using translation invariance, it follows from (4.9) that Ppc med↔ med+ x ∗ m in (med+ Λ(m)) ≥ c(d) md−1.

Note that med+ Λ(m) ⊂ H+(0), med+ x∗m ∈ ∂H+(0) and med ∈ H+(m).

We thus get that, see also Figure 4.4, Ppc med+ x

∗

m ↔ H+(m) in H+(0) ≥

c(d) md−1.

The final assertion now follows from translation invariance.

With this, we are ready to show the final result, stated in the following lemma.

Lemma 4.8. Let d ≥ 2 and define η = 2d2 + 4d − 4. Suppose that (P) does not hold, then there exists a C(d) > 0 such that for all n ∈ N we have Ppc(two-arms(0, n)) ≥ C(d)n

−η_.

Proof. Let d ≥ 2 and assume ‘not (P)’. Recall that Bd(m, n) = [0, n]d−1×

[0, m] and LRd(m, n) = {[0, n]d−1× {0} ↔ [0, n]d−1× {m} in Bd(m, n)}.

Since (P) fails to hold, there exists an > 0 and an N ∈ N such that for all n ≥ N we have (using rotation invariance)

Ppc(LRd(10n, 30n)

c

) ≥ > 0 or Ppc(two-arms(0, n)) > n

−ν _{≥ n}−η_.

Fix such and N . Let n ≥ N . If n is such that Ppc(two-arms(0, n)) ≥ n

−η_,

then we are done. Hence, we assume that Ppc(LRd(10n, 30n)

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Figure 4.4: For d = 2, drawn here is the event med↔ med+ x∗m in med+

Λ(m). Light gray is the set H+(0) \ H+(m).

The event LRd(10n, 30n)c implies in particular that there are no open

paths connecting the faces [0, 30n]d−1_{× {10n} and [0, 30n]}d−1_{× {0} within}

the box Bd(10n, 30n). By translation invariance, we thus get that

Ppc ( [−15n, 15n]d−1× {−5n} ↔ [−15n, 15n]d−1× {5n} in [−15n, 15n]d−1× [−5n, 5n] )c! ≥ . (4.10)

Next, we note that if 5ned ↔ H+(15n) in H+(5n), then in particular

5ned ↔ ∂Λ(15n) in H+(5n), see also Figure 4.5. From Lemma 4.7 and

using symmetry, we obtain that for each σ ∈ {−, +}

Ppc σ5ned↔ ∂Λ(15n) in H

σ_{(5n) ≥} c(d)

(10n)d−1, (4.11)

where c(d) > 0 is as in Lemma 4.7.

We then note that if the faces [−15n, 15n]d−1×{−5n} and [−15n, 15n]d−1_×

{5n} are not connected within the ‘box’ [−15n, 15n]d−1 _{× [−5n, 5n] and}

σ5ned ↔ ∂Λ(15n) in Hσ(5n) occurs for each σ ∈ {−, +}, then the event

A2(5n, 15n) occurs. See also Figure 4.6.

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Figure 4.5: For d = 2, drawn here is the event 5ned↔ H+(15n) in H+(5n).

Light gray is the area H+(5n) \ H+(15n).

we get Ppc(A2(5n, 15n)) ≥ Ppc(LRd(10n, 30n) c )Ppc 5ned↔ ∂Λ(15n) in H +_(5n)2 ≥ c(d) n2d−2,

for some updated c(d) > 0. From Proposition 3.9 we then obtain that Ppc(two-arms(0, n)) ≥ c(d) 1 n2d−2 1 n2d2_+2d−2 ≥ c(d) 1 n2d2_+4d−4,

for some again updated c(d) > 0.

We may assume w.l.o.g. that c(d) ≤ 1, which shows that for all n ≥ N we have

Ppc(two-arms(0, n)) ≥

c(d) n4d2_+4d−4.

We are left to update c(d) such that it holds also for all n ≤ N , which shows the desired result (hence also proves Theorem 4.1).

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Figure 4.6: Drawn for d = 2 the events 5ned↔ ∂Λ(15n) in H+(5n) (green),

−5ned↔ ∂Λ(15n) in H−(5n) (green) and the ‘blocking surface’ (red)

indi-cating that there are no open path from the faces [−15n, 15n]d−1× {−5n} to [−15n, 15n]d−1× {5n} staying inside [−15n, 15n]d−1_{× [−5n, 5n]. In gray}

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4.3 Absence of percolation when the two-arms

ex-ponent is too large

Of course, Theorem 4.1 implies the following statement: if there exists a ξ > η = 2d2 + 4d − 4 such that Ppc(two-arms(0, n)) ≤ n

−ξ _{for infinitely}

many n ∈ N, then θ(pc) = 0. We will show a stronger result.

Theorem 4.9. Let d ≥ 2. If there exists a ξ > 2d2+ 2d − 2 such that for all N ∈ N, there exists an n ≥ N such that

Ppc(two-arms(0, n)) ≤ n

−ξ_,

then θ(pc) = 0.

Cerf presumably has a stronger result, see remark 4.2.

Fix ξ > 2d2 + 2d − 2. Recall that Kn = {x ∈ Λ(5n) : x is open, x ↔

∂Λ(15n)} and In = {x ∈ Λ(15n) : x is open, x ↔ Kn in Λ(15n)}. Recall

definition 4.3 of a good box.

Assume that θ(pc) > 0 and for each N ∈ N, there exists an n ≥ N

such that Ppc(two-arms(0, n)) ≤ n

−ξ_{. We will show that a box is good}

with high probability and hence work up to the point (4.5). From this point, all remaining arguments work because the assumption on the two-arms probability implies lim sup_n→∞Pp(A2(5n, 15n)) = 0 by Proposition

3.9 and Lemma 3.8. It follows that θ_H(pc) > 0. Since this does not occur by

Barsky, Grimmett and Newman [3] (see also Theorem 2.8), we can negate the statement to get

θ(pc) = 0 or ∃N ∈ N, ∀n ≥ N : Ppc(two-arms(0, n)) ≥ n

−ξ_.

By assumption on the two-arms probability, the right-hand side does not occur and we obtain the final result θ(pc) = 0. We thus only have to show

that a box is good with arbitrary large probability. The following lemma is devoted to this.

Lemma 4.10. Let ξ > 2d2 + 2d − 2. Suppose that θ(pc) > 0 and that for

all N ∈ N, there exists an n ≥ N such that Ppc(two-arms(0, n)) ≤ n

−ξ_{. We}

have that for each > 0, there exists an n ∈ N such that Ppc(Λ(15n) is good) > 1 − .

Proof. Fix ξ > 2d2+ 2d − 2. We follow the ideas of Grimmett [14, Section 7.4]. It follows that |K_n| = X x∈Λ(5n) 1x open,x↔∂Λ(15n)≥ X x∈Λ(5n) 1x↔∞.

Bounds on the two-arms probability for percolation on Z^d

University of Amsterdam