A short proof of the phase transition for the vacant set of random interlacements

The vacant set of random interlacements at level $u>0$, introduced in arXiv:0704.2560, is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories, where $u$ is a parameter controlling the density of the cloud. It was proved in arXiv:0704.2560 and arXiv:0808.3344 that for any $d \geq 3$ there exists a positive and finite threshold $u_*$ such that if $uu_*$ then the vacant set does not percolate. We give an elementary proof of these facts. Our method also gives simple upper and lower bounds on the value of $u_*$ for any $d \geq 3$.


Introduction
The model of random interlacements was introduced in [8]. The interlacement I u at level u > 0 is a random subset of Z d , d ≥ 3 that arises as the local limit as N → ∞ of the range of the first ⌊uN d ⌋ steps of a simple random walk on the discrete torus (Z/N Z) d , d ≥ 3, see [14]. The law of I u is characterized by P[I u ∩ K = ∅] = e −u·cap(K) , for any finite K ⊆ Z d , (1.1) where cap(K) denotes the discrete capacity of K, see (2.5). The vacant set of random interlacements V u at level u is defined as the complement of I u at level u: By [8, (1.68)] the correlations of V u decay polynomially for any u > 0: One is interested in the connectivity properties of the subgraphs of the nearest neighbour lattice Z d spanned by the above random sets. For any u > 0, I u is a P-a.s. connected random subset of Z d (see [8, (2.21)]), but V u exhibits percolation phase transition: there exists u * ∈ (0, ∞) such that (i) for any u > u * , P-a.s. all connected components of V u are finite, and (ii) for any u < u * , P-a.s. V u contains an infinite connected component.
The fact that u * < ∞ was proved in [8,Section 3], and the positivity of u * was established in [8,Section 4] when d ≥ 7, and later in [6] for all d ≥ 3.
There is no reason to believe that an exact formula for the value of the critical threshold u * = u * (d) exists. However, it is proved in [9,10] that lim d→∞ u * (d) ln(d) = 1, (1.4) in agreement with the principal asymptotic behaviour of the critical threshold of random interlacements on 2d-regular trees, which is explicitly computed in [12,Proposition 5.2].
The existing proofs of 0 < u * and u * < +∞ on Z d are rather involved and it seems hard to extract quantitative information about the value of u * in low dimensions from them. The aim of this paper is to provide explicit upper and lower bounds on the value of u * = u * (d) for any d ≥ 3.
For any d ≥ 3 let us denote by 0 < c g = c g (d) and C g = C g (d) < +∞ the best constants such that the inequalities hold, where |·| denotes the ℓ ∞ -norm on Z d and g(·, ·) denotes the Green function of simple random walk on Z d , see (2.3). The positivity of c g and the finiteness of C g follow from [4,Theorem 1.5.4]. and (1.8) The bounds (1.6) are not at all sharp, especially if we compare them with (1.4). This shortcoming of Theorem 1.1 is counterbalanced by the fact that its proof is very simple. In particular, our self-contained proof does not use the "sprinkling" technique and decoupling inequalities usually applied in order to overcome the long-range correlations (1.3) present in the model. The proof of u * (d) > 0 for d ≥ 7 in [8, Section 4] does not use "sprinkling", but the proof of u * < +∞ for any d ≥ 3 in [8, Section 3] and the proof of u * (d) > 0 for 3 ≤ d ≤ 7 in [6] does. Various forms of decoupling inequalities have been subsequently developed to study the connectivity properties of V u in the subcritical [5,7,11] and supercritical [2,13] phases. These techniques are very useful once they are available, but the elementary proof of our paper seems to be easier to adapt to other percolation models with long-range correlations, e.g., branching interlacements [1].
Let us briefly describe the idea of the proof of Theorem 1.1. We employ multi-scale renormalization. In order to prove u * < +∞ we show that if V u crosses an annulus at scale L n = 6 n then this vacant crossing contains a set X T of 2 n vertices which arises as the image of leaves under an embedding T of the dyadic tree of depth n (this construction already appears in [11]). The number of possible embeddings is less than C 2 n d (c.f. (1.7)), so we only need to show that cap(X T ) ≍ 2 n if we want to use (1.1) to to show that crossing of the annulus by V u is unlikely when u is big enough. This is indeed the case, because by construction the embedding T is "spread-out on all scales", thus the cardinality and the capacity of X T are comparable. In order to prove u * > 0, we restrict our attention to a plane inside Z d . By planar duality we only need to show that a * -connected crossing of a planar annulus at scale L n = L 0 · 6 n by I u is unlikely. We show that such a crossing must intersect 2 n "frames", where each frame is the union of four "sticks" of length 2L 0 − 1. Such a collection of frames again arises from a spread-out embedding of the the dyadic tree of depth n. We use that I u can be written as the union of the ranges of a Poissonian cloud of independent random walks and the fact that random walks tend to avoid sticks if L 0 is large enough (c.f. (1.8)) to arrive at a large deviation estimate on the probability that the number of frames that intersect I u is 2 n which is strong enough to beat the combinatorial complexity term C 2 n 2 . This stick-based approach to u * > 0 is already present in [6, Section 3] and our large deviation estimate resembles the one in the proof of [8,Theorem 2.4].
The rest of this paper is organized as follows. In Section 2 we introduce further notation and recall some useful facts related to the notion of capacity and random interlacements. In Section 3 we define the notion of a proper embedding of a dyadic tree into Z d and derive some facts about such embeddings. In Sections 4 and 5 we prove the upper and lower bounds on u * stated in Theorem 1.1.

Preliminaries
For a set K, we denote by |K| its cardinality. We denote by the law of simple random walk with initial distribution m and by E m the corresponding expectation. The Green function of simple random walk on Z d is defined by Let us denote by {X} ⊆ Z d the range of the random walk:

Potential theory
The total mass of the equilibrium measure is called the capacity of K: One defines the normalized equilibrium measure e K (·) of K by Let us now collect some facts about capacity that we will use in the sequel. The proofs of the properties (2.7)-(2.10) below can be found in, e.g., [3,Section 1.3].
For any x ∈ Z d and any K ⊂⊂ Z d , . (2.10) Let us denote by F the plane For any y ∈ F and L ≥ 1 let us define the frame L y ⊆ F by L y The next lemma gives an explicit upper bound on the capacity of a frame. Recall the notion of c g from (1.5).

Constructive definition of random interlacements
The definition of the interlacement I u at level u by the formula (1.1) is short, but it is not constructive. The construction of [8, Section 1] involves a Poisson point process with intensity measure u·ν, where ν is a sigma-finite measure on the space of equivalence classes of doubly infinite trajectories modulo time shift. The union of the ranges of trajectories which are contained in the support of this Poisson point process is denoted by I u , and this random subset of Z d indeed satisfies (1.1).
We will not use the full definition of random interlacements, only a corollary of it, which allows one to construct a set with the same law as I u ∩ K for any K ⊂⊂ Z d .
Claim 2.2. Let d ≥ 3, K ⊂⊂ Z d , N K be a Poisson random variable with parameter u · cap(K), and (X j ) j≥1 i.i.d. simple random walks with distribution P e K and independent from N K . Then K ∩ ∪ N K j=1 {X j } has the same distribution as I u ∩ K.

Renormalization
3. for all 0 ≤ k < n and m ∈ T (k) we have We denote by Λ n,x the set of proper embeddings of T n into Z d with root at x. Proof. If n ≥ 1, x ∈ L n and T ∈ Λ n,x , we denote by T 1 and T 2 the two embeddings of T n−1 which arise from T as the embeddings of the descendants of the two children of the root, i.e., for any 0 ≤ k ≤ n − 1 and m = (ξ 1 , . . . , ξ k ) ∈ T (k) let T ξ (m) = T (ξ, ξ 1 , ξ 2 , . . . , ξ k ) for ξ ∈ {1, 2}. By Definition 3.1 we have T ξ ∈ Λ n−1,T (ξ) for ξ ∈ {1, 2}, thus we obtain (3.4) by induction on n: where in ( * ) we used the induction hypothesis.
Recall the notion of S(x, R) from (2.1) and note that S(x, 0) = {x}. Proof. We will prove that (3.5) implies that there exists T ∈ Λ n,x such that for all 0 ≤ k ≤ n we have We will construct such a T ∈ Λ n,x by induction on k. By T (∅) = x we see that the case k = 0 of (3.7) is just (3.5). Assuming that (3.7) holds for some 0 ≤ k ≤ n − 1 we now show that it also holds for k + 1. If m ∈ T (k) then our induction hypothesis (3.7) and the fact that γ is a * -connected path imply We have thus constructed the embedding T up to depth k + 1 so that Definition 3.1 is satisfied up to depth k + 1 and (3.7) also holds for k + 1. Therefore by induction we have constructed T ∈ Λ n,x such that (3.7) holds for all 0 ≤ k ≤ n, which implies (3.6). The proof of Lemma 3.3 is complete.

Upper bound on u *
Let us choose L 0 = 1 in (3.2). For n ≥ 1 let us denote by A u n the event there exists a nearest neighbour path in V u that connects S(0, L n − 1) to S(0, 2L n ) .
Recall the definitions of C g from (1.5) and C d from (1.7).
Proposition 4.1. For any d ≥ 3 and there exists q = q(d, u) ∈ (0, 1) such that for any n ≥ 1 we have Corollary 4.2. Proposition 4.1 implies the upper bound of Theorem 1.1, as we now explain. Let us denote by A u n the event that there exists a nearest neighbour path in V u that connects S(0, L n −1) to infinity and by A u ∞ the event that V u has an infinite connected component. If (4.1) holds, then where ( * ) holds by monotone convergence. Therefore we have u * ≤ 5 2 C g ln(C d ). Proof of Proposition 4.1. For any n ≥ 1 and T ∈ Λ n,0 we denote Noting that S(T (m), L 0 − 1) = S(T (m), 0) = {T (m)} for any m ∈ T (n) and that every nearest neighbour path is also a * -connected path we can apply Lemma 3.3 to infer In order to finish the proof of Proposition 4.1 we only need to show that for any T ∈ Λ n,0 we have cap(X T ) ≥ 2 5 We will show (4.4) using (2.10). For any T ∈ Λ n,0 and any m ∈ T (n) we have Now (4.4) follows from (2.10), (4.5) and the fact that |X T | = 2 n . The proof of Proposition 4.1 is complete.

Lower bound on u *
Let us choose L 0 according to (1.8) in (3.2). Recall the notion of the plane F from (2.11). For n ≥ 1 and x ∈ L n ∩ F let us denote by B u n,x the event B u n,x = there exists a * -connected path in I u ∩ F that connects S(x, L n − 1) to S(x, 2L n ) .
Recall the definitions of c g , C g from (1.5) and C d from (1.7).
Proposition 5.1. For any d ≥ 3 and for any n ≥ 1 and x ∈ L n ∩ F we have Corollary 5.2. Proposition 5.1 implies the lower bound of Theorem 1.1, as we now explain. Let us denote by A u n the event that there exists a nearest neighbour path in V u ∩ F that connects S(0, L n ) to infinity and by A u ∞ the event that V u ∩ F has an infinite connected component. By planar duality the event ( A u n ) c is equal to the event that there exists a * -connected path in I u ∩ F that surrounds S(0, L n − 1), thus if (5.1) holds, then which in turn implies P[ A u ∞ ] = lim n→∞ P[ A u n ] = 1. Therefore we have u * ≥ cg L 0 1 C 2 2 −(d+5) . Proof of Proposition 5.1. We say that T : T n → F is a proper embedding of the dyadic tree T n with root at x ∈ L n ∩ F into F if T ∈ Λ n,x (see Definition 3.1). We denote by Λ F n,x the set of proper embeddings of T n into F . For any y ∈ L 0 ∩ F let us define the frame y ⊆ F by y (2.12) For any n ≥ 1, x ∈ L n ∩ F and T ∈ Λ F n,x let us denote by We start the proof of Proposition 5.1 by an application of Lemma 3.3: where in ( * ) we used Lemma 3.2 to infer |Λ F n,x | ≤ C 2 n 2 . In order to bound the probability on the right-hand side of (5.4) let us fix some T ∈ Λ F n,x , recall the constructive definition of random interlacements from Claim 2.2 and denote the probability underlying the random objects (i.e., N K and (X j ) j≥1 ) introduced in that claim by P when K = X T . For a simple random walk X let us denote by the number of frames of form T (m) , m ∈ T (n) that X visits. We can bound The bound (5.7) together with the strong Markov property of simple random walk imply that P e K [N (X) ≥ k] ≤ p k−1 for any k ≥ 1. In other words, N (X) is stochastically dominated by a geometric random variable with parameter 1 − p, which implies E e K z N (X) ≤ (1−p)z 1−pz for any 1 ≤ z < 1 p . Recalling from Claim 2.2 that N K is Poisson with parameter u · cap(K) = u · cap(X T ), for any 1 ≤ z < 1 p we obtain E z N K j=1 N (X j ) = exp u · cap(X T ) E e K z N (X) − 1 ≤ exp u · cap(X T ) z − 1 1 − pz .
We can thus apply the exponential Chebyshev's inequality with z = 1 2p to bound P B u n,x This completes the proof of Proposition 5.1.