Some Rigorous Results on the Phase Transition of Finitary Random Interlacement

In this paper, we show several rigorous results on the phase transition of Finitary Random Interlacement (FRI). For the high intensity regime, we show the existence of a critical fiber length, and give the exact asymptotic of it as intensity goes to infinity. At the same time, our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity.


Introduction
The model of finitary random interlacements (FRI) was first introduced by Bowen [1] to solve a special case of the Gaboriau-Lyons problem. Intuitively speaking, FRI (denoted by F I u,T ) can be seen a "cloud of twisted yarn" composed of finite " fibers" on Z d . The fibers within this system each form a geometrically killed simple random walk, where the starting points are sampled according to a Poisson point process with intensity measure proportional to system intensity u and inversely proportional to T + 1. At the same time, the expected length of each fiber is given by T [1,2,3,8]. See Section 2 for precise definitions.
In [1], Bowen proved that as T → +∞ , F I u,T converges under weak star topology to the celebrated random interlacements (RI) introduced by Sznitman [10], with intensity u. FRI can also be seen as a finitary version of RI, in the sense that it can be defined as massive interlacements on a graph equipped with killing measure (see Chapter V of [7]).
On the other hand, unlike the classic RI, whose trajectories always form an a.s. connected network [4,10], the collection of edges traversed by FRI on Z d has exhibited a non-trivial percolative phase transition [8]. And since there are now two parameters to play with, one may characterize such transition from either of the following viewpoints: (1) One may first fix the intensity u and examine the evolution of FRI with respect to T . This is also the setting Bowen used in his first paper [1]. In addition to the convergence when T → ∞, he also proved that the FRI on non-amenable graphs will a.s. have infinite connected cluster(s) for all sufficiently large T . He proposed same question for FRI on Z d , which was affirmatively answered in [8] by proving the existence of the following phase transition with respect to T : For any d ≥ 3 and u > 0, there is T 0 , T 1 ∈ (0, ∞), such that F I u,T a.s. has no infinite 1 cluster for all T < T 0 , and a.s. has a unique infinite cluster for all T > T 1 (see Theorem 1,2 in [8] for details). The geometric properties for the infinite cluster, such as local uniqueness and order of chemical distance, for sufficiently large T was later obtained in [2]. However, as mentioned in Section 1.1 of [8], the uniqueness of phase transition and the existence/uniqueness of a critical fiber length T c remain open. Unlike the case for RI, this turns out to be a non-trivial problem, since it was recently proved that there is no global stochastic monotonicity with respect to T for F I u,T , as shown in Theorem 1, [3]. On the other hand, numerical tests in Section 5 of [3] provide evidences on the existence and uniqueness of T c , which was also conjectured in Conjecture 5 [3] to be aymptotically an inverse linear function with respect to u.
(2) As in [7], one may also fix T and examine the evolution of FRI with respect to its intensity u. Note that FRI is by definition monotone with respect to u. So the question of interest here is whether there is always a non-trivial phase transition. I.e., for a fixed T , we want to show FRI does not percolate for all sufficiently small u and percolate for all sufficiently large u. [7] proved this for all T small enough, and conjectured it can be extended to all T ∈ (0, ∞).
In this paper, we prove that, despite lacking global monotonicity, FRI is stochastically increasing with respect to T for all T ∈ (0, 1), which implies the existence and uniqueness of T c for all sufficiently large u. Meanwhile, we also show that the upper bound of T c found by [3,7] in the high intensity regime is actually sharp, and give an exact asymptotic of T c as u → ∞. Moreover, for the low intensity regime, we prove a polynomial lower bound for the phase diagram, which at the same time proves the conjecture on the global existence of a non-trivial phase transition with respect to u. Our proofs are largely based on the "decoupling" methods first invented in [11].
This paper is organized as follows. In Section 2 we introduce precise definitions of FRI together with some necessary notations and preliminaries. We state our main results in Section 3. Local monotonicity is shown in Section 4. And at last we estimate asymptotic of critical values in Section 5.

Notations and preliminaries
Some basic notations: In this paper, we denote the l ∞ distance and Euclidean distance by | · | and | · | 2 respectively. We also denote the undirected edge set of Z d by L d (i.e. L d := {x, y} : x, y ∈ Z d , |x − y| 2 = 1 ). For any sets A, B ⊂ Z d , the l ∞ distance between them is defined as d(A, B) := min{|x − y| : x ∈ A, y ∈ B}. For finite subset D ⊂ Z d , let ∂D := {x ∈ D : ∃y ∈ Z d \ D such that {x, y} ∈ L d } be its inner boundary and |D| be the cardinality of D, without casusing further confusion.
Connection between two sets: For sets A, B ⊂ Z d and a collection of edges in L d denoted by E, we say A and B are connected by E (written by A there exists a sequence of vertices (x 0 , ..., x n ) such that x 0 ∈ A, x n ∈ B and that for any Statements about constants: we will use c, c 1 , c 2 , ... as local constants ("local" means their values may vary according to contexts) and C, C 1 , C 2 , ... as global constants ("global" means constants will keep their values in the whole paper).
Random walks and relative stopping times: We denote the law of simple random walks starting from x on Z d by P x and the law of geometrically killed simple random walks starting from x with killing rate 1 T +1 at each step by P (T ) x . For a random walk {X i } ∞ n=0 and A ⊂ Z d , define the hitting time and entrance time as Capacity with killing measure: For any K ⊂ Z d , the escaping probability is defined as Es In particular, we have Definitions of FRI: According to [8], FRI has two equivalent definitions. Denote the set of all finite nearest-neighbor paths on Z d by Definition 1. For 0 < u, T < ∞, finitary random interlacements F I u,T is the Poisson point process with intensity measure u * v (T ) . We denote the law of F I u,T by P u,T .
∼ P ois( 2du T +1 ). For each site x ∈ Z d , start N x independent geometrically killed simple random walks with law P (T ) x . Let F I u,T be the point measure on W [0,∞) composed of all the trajectories above starting from all x ∈ Z d .
With a slight abuse of notations, for η ∈ W [0,∞) , we may write η ∈ F I u,T if F I u,T (η) = 1. In this paper, F I u,T is also regarded as a bond percolation model on L d . I.e. we say an edge e ∈ L d is open iff there exists a path in F I u,T containing e and then write F I u,T (e) = 1 (otherwise, F I u,T (e) = 0). For the simplicity of notations, for any A, B ⊂ Z d , we denote We say F I u,T percolates iff there exists an infinite connected cluster composed of open edges. It has been proved in Theorem 2, [3] that F I u,T contains at most one open infinite connected cluster.
FRI on a finite set: Let K be a finite subset of Z d . For each path η i in F I u,T , we denote the part of η i after intersecting K byη K i . Precisely, for i δηK i has the same law as a Poisson point process with intensity measure u * x∈K Es x . As a direct corollary, the number of paths intersecting K in F I u,T is a Poisson random variable with parameter u * cap (T ) (K).
Independence in FRI: Now we give the definition of critical values of FRI: By Theorem 1 and 2 of [8], it has been peroved that for all d ≥ 3,

Main results
In our first result, we show that for all d ≥ 3 and u > 0, although F I u,T does not enjoy monotonicity for larger T 's (see Theorem 1, [3]), it is indeed stochastically increasing with respect to T for all T ∈ (0, 1]. Theorem 1. For any u > 0, 0 < T 1 < T 2 < ∞ such that T 1 * T 2 ≤ 1, then F I u,T 2 stochasitcally dominates F I u,T 1 . I.e. there is a coupling between F I u,T 1 and F I u,T 2 such that almost surely for any edge e ∈ L d , F I u,T 1 (e) ≤ F I u,T 2 (e).
Combining Theorem 1 together with Theorem 3 (iv) and Proposition 2, [3], one can now have the existence and uniqueness of a critical fiber length for all sufficiently large u.
. So we have F I u,T percolates a.s. for all T > T c , and does not percolates a.s. for all T < T c .
Our next result provides the exact asymptotic of T c as u → ∞, which gives an affirmative answer to Part 2, Conjecture 5, [3].
where p c d is the critical value of Bernoulli bond percolation on L d . Remark 2. In Theorem 4.2 [3] and on Page 263 [7], it has been shown that (3.2) lim sup So here we only need to prove a sharp lower bound for the subcritical phase (a partial result on the asymptotic order was given in Proposition 2, [3]).
Next we estimate the subcritical phase in the low intensity regime.
Theorem 3. For any d ≥ 3 and δ > 0, there exist constants 0 < U 0 (d, δ) < 1 and C 1 (d, δ) > 0 such that: for any 0 < u ≤ U 0 , Combining Theorem 3 and the supercritical estimates obtained in (v) of Theorem 3, [3], the following result can be summarized under log-log plot on the bounds we have about the phase transition.
When d ≥ 4, Finally, Theorem 2 together with Theorem 3 also give an affirmative answer to the conjecture posed on Page 263, [7] on the global existence of a non-trivial phase transition with respect to u: Theorem 4. For all d ≥ 3 and T ∈ (0, ∞), there is a u c = u c (d, T ) ∈ (0, ∞) such that F I u,T percolates a.s. for all u ∈ (u c , ∞), and does not percolate a.s. for all u ∈ (0, u c ).

Local stochastic monotoncity
Before giving the proof of Theorem 1, we introduce a new approach to construct F I u,T . Denote the collection of all directed nearest-neighbor edges bŷ and X y,k · k∈N + ,y∈Z d be a sequence of independent random walks with law P Proof. For any x ∈ Z d and x → y ∈L d , by Definition 2 and property of Poisson point process, we know the number of paths starting from x with length ≥ 1 and the first step x → y is a Poisson variable with parameter T T +1 * 1 2d * 2du T +1 = uT (T +1) 2 . In addition, by the memoryless property of the geometric distribution, the remaining part of each path after the first step removed is still a geometrically killed random walk with killing rate 1 T +1 . Then Lemma 4.1 follows. Now we give the proof of Theorem 1 as follows: Proof of Theorem 1: When T 1 < T 2 , 0 < T 1 * T 2 < 1, we define three independent sequences of random variables: Meanwhile, it's elementary to construct Note that for any x → y, N (1) x→y + N

Note that for any
Finally, by comparing the LHS's of (4.1) and (4.2), we get the stochasitc domination in Theorem 1. 6

Asymptotic of critical values
In this section, we prove Theorem 2 and Theorem 3. Before presenting the proof, we need to first introduce some notations according to [9] in order to implement the renormalization arguement.
(1) Let L 0 and l 0 be positive integers. For n ≥ 1, let L n = l n 0 * L 0 and L n = L n * Z d .
(5.1) By (2.8) of [9], one has Based on these settings, we can do decompositions on the events and then estimate their probabilities by choosing proper L 0 and l 0 . Roughly speaking, we need to select L 0 to control the 0−level event and select l 0 to guarantee the "almost indepedence" between trajectories in different boxes, according to u and T . Now we give the proof of Theorem 2.

Proof of Theorem 2.
Recall that in Theorem 3.(iii) of [3] and on Page 263 of [7], it has been shown that Now it's sufficient to prove that: for any ǫ > 0, there exists U ′ (d, ǫ) > 0 such that for any u > U ′ and T > 0 satisfying F I u,T does not percolate, from which we deduce Assume u is sufficiently large while T satisfies (5.4). Take L 0 = ⌊u 1 2d ⌋ > 10 and l 0 = 10. Let B * 0,0 = {y ∈ Z d : d({y}, B 0,0 ) ≤ 1} and recall the notationη K in Section 2, then we have For any n ≥ 0 and x ∈ L n , we write that Recall the notation Λ n,x in (5.1). For any T ∈ Λ n,x , like (2.13) of [9], we write (5.10) Similar to (2.14) of [9], we have Here we need a decoupling inequality, which is parallel to Lemma 5.4, [2]. Let If F n,x does not occur, there must exist a path η ∈ F I u,T such that η ∩∂B n,x = ∅ and η ∩ ∂ B n,x = ∅. Recalling the construction of FRI on a finite set in Section 2 (here we take K = ∂B n,x ∪ ∂ B n,x ), the number of such paths is a Poisson random variable with parameter (5.12) By l 0 = 10 and (5.12), when u is sufficiently large, we have: for any n ≥ 0, Assume that T ∩ I n−1 = {(n − 1, y 1 ), (n − 1, y 2 )}, where y 1 , y 2 ∈ L n−1 . Since L 0 > 10,B n−1,y 1 ∩B n−1,y 2 = ∅. We denote that T i = {(m, y) ∈ T : y ∈ B n−1,y i }, i ∈ {1, 2}. We also write (5.14) Note thatÂ T 1 andÂ T 2 are independent and for i ∈ {1, 2},Â T i ⊂ A T i . In addition, if F n−1,y 1 and F n−1,y 2 both occur, eventsÂ T andÂ T 1 ∩Â T 2 will be equivalent.
In conclusion, Then we get Theorem 2 by (5.18) and Corollary 1.

Proof of Theorem 3:
Before we give the proof of Theorem 3, we need an estimate on the diameter of the range of geometrically killed random walks.  Proof. By Theorem 1.5.1 of [6], for any positive integer m, we have By Theorem 1, for any fixed u > 0, F I u,T is stochastically increasing on T ∈ (0, 1]. Thus, it's sufficient to confirm: for d ≥ 3 and ǫ > 0, there existsc(d, ǫ) > 0 and U ′′ (d, ǫ) > 0 such that for any u < U ′′ and T ≥ 1 satisfying F I u,T does not percolate. In fact, the result proved here is sightly stronger than Theorem 3 and the RHS of (5.21) can be replaced by a polynomial of T to make the proof a little shorter.
We use the same approach as in the proof of Theorem 2 with different L 0 and l 0 . Precisely, we set L 0 = 10, l 0 = ⌊ c ′ T * (log(T + 1)) 3+ǫ 0.5 ⌋, where c ′ will be determined later. Note that the number of paths intersecting B 0,0 in F I u,T is a Poisson random variable with parameter u * cap (T ) (B 0,0 ). Therefore,