Random interlacements on Galton-Watson Trees

We study the critical parameter u^{*} of random interlacements percolation (introduced by A.S Sznitman in arXiv:0704.2560) on a Galton-Watson tree conditioned on the non-extinction event. Starting from the previous work of A. Teixeira in arXiv:0907.0316, we show that, for a given law of a Galton-Watson tree, the value of this parameter is a.s. constant and non-trivial. We also characterize this value as the solution of a certain equation.


Introduction
The aim of this note is to the study random interlacements model on a Galton-Watson tree. We will mainly be interested in the critical parameter of the model. In particular we want to understand whether this parameter is non trivial (that is different from 0 and ∞), whether it is random and how it depends on the law of the Galton-Watson tree. Our main theorem answers all of these questions and even goes further by characterizing the critical parameter as the solution of a certain equation. The random interlacements model was recently introduced on d , d ≥ 3, by A.S. Sznitman in [Szn10] and generalised to arbitrary transient graphs by A. Teixeira in [Tei09]. It is a special dependent site-percolation model where the set of occupied vertices on a transient graph (G, ) is constructed as the trace left on G by a Poisson point process on the space of doubly infinite trajectories modulo time shift. The density of the set is determined by a parameter u > 0 which comes as a multiplicative parameter of the intensity measure of the Poisson point process. In this paper, we will not need the complete construction of the random interlacements percolation. For our purposes it will be sufficient to know that the law Q G u of the vacant set = G \ of the random interlacements at level u is characterized by The critical parameter u * G of random interlacements on G is defined as: s. all connected components of are finite . (1.2) In this article we take the graph G to be a Galton-Watson rooted tree T defined on a probability space (Ω, , ), conditioned on non-extinction. We denote by ∅ the root of the tree, ρ i i≥0 the offspring distribution of the non-conditioned Galton-Watson process, f its generating function and q the probability of the extinction event.¯ stands for the conditional law of the Galton-Watson tree on the non-extinction event and¯ for the corresponding conditional expectation. We assume that T is supercritical that is If (A0) is satisfied, then¯ is well defined and T is¯ -a.s. transient, see for example Proposition 3.5 and Corollary 5.10 of [LP]. Thus the random interlacements are defined on T and in particular Q T u is¯ -a.s. well-defined. To state our principal theorem we introduce T the sub-tree of T composed of the vertices which have infinite descendence. We recall the Harris decomposition (cf. Proposition 5.26 in [LP]): under¯ , T is a Galton-Watson tree with generating function f given by Theorem 1. Let T be a Galton Watson tree with a law satisfying (A0). Then there exists a nonrandom constant u * ∈ (0, ∞) such that Moreover if we denote by χ (u) =¯ e −ucap T ({∅}) the annealed probability that the root is vacant at level u (cf 1.1), then u * is the only solution on (0, ∞) of the equation The main difficulty in proving this theorem is the dependence present in the model. More precisely, for any x ∈ T , the probability that x ∈ is given by The capacity cap T (x) depends on the whole tree T . It is thus not possible to construct the component of the vacant set containing the root of a given tree as a sequence of independent generations, which is a key property when proving an analogous statement for Bernoulli percolation on a Galton-Watson tree. However, it turns out that despite this dependence, it is possible to construct a recurrence relation under the annealed measure¯ ⊗ Q T u for a well chosen quantity related to the size of the cluster at a given point (see (4.11)). This is done in section 4. Using this recurrence relation, it is possible to find an annealed critical parameter u * and show that it is non-trivial. In section 3 we prove that u * T is¯ −a.s. constant and thus that u * T = u * for¯ −a.e. tree. Section 2 introduces some preliminary definitions and recalls some useful results for random interlacements on trees.

Definitions and preliminary results
Let us introduce some notations first. For a given tree T with root ∅ and a vertex x ∈ T \ {∅}, we writex for the closest ancestor of x in T , |x| for its distance to the root, Z x for the number of children of x and deg T (x) for its degree. We denote by T x the sub-tree of T containing x and all its descendants. If T is a tree with root ∅, for every child x of ∅ we will say that T x is a descendant tree of T . For any infinite rooted tree T , we denote by Z x the number of children of x in T . For a tree T and a vertex x of T , we denote by P T x the law of a simple discrete time random walk X n n≥0 started at x. For every set K in T , we use H K to denote the hitting time of the set K defined by We write e T K for the equilibrium measure of K in T and cap T (K) for its total mass, also called capacity of the set K: We also denote x the connected component of containing x. According to Corollary 3.2 of [Tei09], the definition (1.2) of the critical parameter u * T is equivalent to Finally we recall Theorem 5.1 of [Tei09] which identifies the law of the vacant cluster x on a fixed tree T with the law of the vacant set left by inhomogeneous Bernoulli site percolation. This theorem also allows us to compare random interlacements on T and random interlacements on its descendant trees.

3¯ -a.s. constancy of u *
In this section we prove that for a given Galton-Watson tree T satisfying (A0) the critical parameter u * T is¯ -a.s constant. We will use Theorem 2 to prove a zero-one law for the event Q T u | ∅ | = ∞ = 0 . The proof of this zero-one law is based on the following definition and lemma which we learnt in [LP]. We present here its proof for sake of completeness.
Definition 3. We say that a property of a tree is inherited if the two following conditions are satisfied: T has ⇒ all descendent trees of T have . (3.1) All finite trees have . (3. 2) The zero-one law for Galton-Watson trees associated to such properties is: If is an inherited property then, for every Galton-Watson tree process T satisfying (A0), Proof. Let E be the set of trees that have the property and Z ∅ the number of children of the root. Using condition (3.1) we can write Conditionally on Z ∅ , the random trees T x x:|x|=1 are independent and have the same law as T .
Hence the last inequality is equivalent to By assumption (A0) we know that ρ i > 0 for some i ≥ 2. Therefore f is strictly convex on (0, 1) with f q = q and f (1) = which finishes the proof of the lemma.
Proof of (1.4). To prove that u * T is¯ -a.s constant we show first that, for a tree T with root ∅, the property u defined by is inherited. Since every finite tree has u by definition, we just have to prove the statement ∃x ∈ T, |x| = 1 : T x has not u ⇒ T has not u .
(3.7) Let x be a child of the root such that T x = ∞ and T x has not u which can also be written . Using Theorem 2 this means that, conditionally on {x ∈ }, the law of the cluster x ∩ T x under Q T u and the law of the cluster x under Q T x u are the same. In particular we have (3.8) Moreover, we see from (2.5) that h x T (z) = h ∅ T (z) for every z ∈ T x \ {x}. Applying Theorem 2 to the clusters ∅ and x , we see that the law of ∅ ∩ T x under Q T u [.|∅, x ∈ ] is the same as the law of x ∩ T x under Q T u [.|x ∈ ]. In particular we have (3.9) Since on {∅, x ∈ }, ∅ and x are in the same open cluster, we can rewrite (3.8) and (3.9) as which finishes the proof that u is inherited.
We can now apply Theorem 3 and deduce that T has u ∈ {0, 1}. Thus for every s ∈ + , there exists a set A s ⊂ Ω such that¯ A s = 1 and 1 Q T s [| ∅ |=∞]=0 is constant on A s . This yields is constant on A = ∩ s∈ + A s with¯ [A] = 1. But, since u → Q T u | ∅ | = ∞ is decreasing, we also have: It follows directly that u * is -a.s constant.

Characterization of u *
In this section we will show that u * is non-trivial and can be obtained as the root of equation (1.5).
In order to make our calculation more natural we will work with the modified tree T obtained by attaching an additional vertex ∆ to the root of T . This change is legitimate only if T and T have the same critical parameter u * , which is equivalent to To prove (4.1) we observe that by Theorem 2 Since Q T u [∅ ∈ ] > 0 and Q T u [∅ ∈ ] > 0 this is equivalent to so that (4.1) holds and u * T = u * T . If |x| ≥ 1, T x is isomorphic to the tree obtained by attachingx to T x . We will thus identify both trees and write T x for the tree T x ∪ {x}. For every tree T we define the random variable (4.4) The random variables γ T x |x|=1 are related to the random variable χ (T ) := cap T (∅) by The second equality is an easy consequence of definition of the capacity and the third equality follows, using Markov property, from the following computation: We also define γ the Laplace transform of T → γ (T ) under¯ . The recursive structure of Galton-Watson tree implies that the random variables γ T x |x|=1 are i.i.d. We can use this property and formula (1.3) to express relation (4.5) in terms of Laplace transforms. This yields Since f −1 = 1 f • f −1 , this allows us to write (1.5) as = 1 (4.9) and thus f γ (u) = 1. (1.5)' Moreover, f and f being bijective, the uniqueness of the solution is preserved. Thus from now on we will consider (1.5) and (1.5)' as equivalent.
We will now explicit a relation verified by the annealed probability that ∅ is infinite.
Proposition 5. For a Galton-Watson process, r u :=¯ Q T u ∅ = ∞ is the largest root in [0, 1] of the equation Proof. For every vertex x ∈ T , we introduce the notation the relative depth of the cluster containing x. Since We can also write (4.13) According to Theorem 2, we know that conditionally on {∅ ∈ }, under Q T u , ∅ has the same law as a cluster obtained by Bernoulli site-percolation. Hence the random variables (4.14) So that we can rewrite (4.13) as where in the last equation we use the fact that Q T x u -a.s we have the inclusion x ≥ n ⊆ {∅ ∈ }. According to (1.1) and (2.3), we have Using (4.5) and (4.16), (4.15) can be rewritten as We can now finally prove (4.10). If we denote r u n =¯ Q T u ∅ ≥ n , we have The sequence r u n n∈ is decreasing by definition. Therefore it converges and its limit r u = Q T u ∅ = ∞ verifies The function f is strictly convex by (A0). Therefore the function is strictly concave and the equation has at most two roots in [0, 1]. According to (4.8), 0 is always a root. Assume that there exists another root x 0 in (0, 1]. Then using the concavity we have on 0, x 0 . The sequence r u n n∈ is positive and decreasing and verifies (4.18). Thus (4.22) and an easy recurrence shows that ∀n ∈ , r u n ≥ x 0 . Since r u verifies (4.19), r u is a root of (4.21) and r u can only be x 0 in this case. Finally, the tree T has locally finite degree and thus ∅ = ∞ = ∅ = ∞ . This yields is the largest root of (4.10) in [0, 1] which finishes the proof of Proposition 5.
We are now able to deduce non-triviality and (1.5)' from the deterministic study of the roots of the equality (4.10).