Growing in time IDLA cluster is recurrent

We show that Internal Diffusion Limited Aggregation (IDLA) on $\mathbb{Z}^d$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [H], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.


Introduction
This work is concerned with proving recurrence of simple random walk on certain type of growing clusters. Specifically, we consider the Internal Diffusion Limited Aggregation (IDLA [14]) {D n } n∈N on Z d , d ≥ 3 in discrete time, that is, starting from D 0 = {0} consisting of the origin, one iteratively grows the cluster by adding to D n the first vertex v n ∈ Z d \D n that is hit by an independent particle injected from the origin performing discrete-time simple random walk (SRW), and set D n+1 = D n ∪ {v n }, for every n ∈ N. As such, D n are random, finite connected vertex subsets of Z d , monotonically increasing with |D n | = n, where we denote by | · | the cardinality of a set. Being a canonical model of Laplacian growth, IDLA has been much studied, cf. [9,14,10,2,3,12,13] and references therein, with the well-known shape and fluctuation theorems stating, in particular, that for each d ≥ 3, there exists finite constant a = a(d), such that almost surely for all n = ω d r d large enough, we have that B r−a √ log r ⊂ D n ⊂ B r+a √ log r , where ω d denotes the volume of a unit ball in R d and B R = B R ∩ Z d for the origin-centered Euclidean ball B R of radius R. In this article, we view each D n as a subgraph of Z d imposing an edge between any two vertices of Z d -distance 1 within the cluster, and the inclusion above holds in the sense of graphs.
Consider now an (autonomous) simple random walk {X n } n∈N evolving independently on {D n } n∈N starting from X 0 = 0. That is, fixing a realization ω of the IDLA cluster {D ω n } n∈N , at each step n, X n ∈ D ω n chooses uniformly among all vertices in D ω n having Z d -distance 1 from it and passes to X n+1 , before domain changes from D ω n to D ω n+1 .
There is at least one vertex available for X n and at most 2d. Fixing ω, such random walk {X n } n∈N is a time-inhomogeneous Markov chain, albeit at perpetual non-equilibrium, but we are interested in whether the walk is recurrent to the origin, and whether recurrence * Courant Institute of Mathematical Sciences, USA. E-mail: rh138@nyu.edu holds for almost every realization ω of its environment. Indeed, we have studied this problem within a more general set-up in [8] cases to be open is that one needs to rule out the possibility that the evolving IDLA boundary be too rough and traps the random walk, thus delaying its return to the bulk. This is not automatically ruled out by the limiting shape. Besides, in the timeinhomogeneous setting one loses some robust tools such as commute time identity from electrical networks.
It turns out that the issue can be resolved by understanding the isoperimetry property of a typical large cluster. In a previous work [11], we give sufficient conditions for recurrence (and transience) of SRW evolving independently on sequences of growing subgraphs of a pre-given infinite graph, in terms of the behaviors of the Cheeger (i.e. isoperimetric) constant of each subgraph in that sequence. When certain sufficient condition is satisfied, see (2.6), the heat kernel of X n at its starting point can be bounded below by c/|D n | at time n, for some absolute constant c > 0. This yields corresponding lower bounds on the expected occupation time (i.e. Green's function), upon integrating the heat kernel. The framework includes as special case the IDLA cluster on Z d , as soon as the cluster is large enough. In this work, we show that the isoperimetry of large IDLA cluster is near optimal, namely it differs from the isoperimetry of a lattice ball by at most a logarithmic correction, see Theorem 2.3, which implies recurrence of the random walk, see Corollary 2.7. To derive the former, we rely heavily on the known limit shape and fluctuation bounds from circularity. The relation between isoperimetry and heat kernel or mixing time estimates on fixed graphs is well understood, cf. [4,18,20]. Even in random environments such as supercritical percolation clusters, a key step in deriving the latter estimates is to understand the precise isoperimetric inequality satisfied by the cluster at large scales, see [5,17,6,19]. Here we see something similar, though in a dynamic setting.

Framework and main results
We work first in discrete time, denoting by {D n } n∈N the growing IDLA cluster on Z d . We recall its growth mechanism: starting at D 0 = {0}, the random cluster grows by adding one vertex per integer time, chosen by an independent particle, denoted Y n , doing simple random walk from the origin till visiting the first vertex v n outside of cluster, then we set D n+1 = D n ∪ {v n }. Hence, |D n | = n. While respecting its growth mechanism, we omit particle Y n 's travel time and consider it settled on Z d \D n instantaneously upon injected, for every n ∈ N. Each D n is then viewed as a subgraph of the lattice graph Z d , by imposing an edge between any x, y ∈ D n of Z d -distance 1, which we denote x ∼ y.
For d = 2, the same holds with log r replacing √ log r in S n .
We also require some notation for general graphs. To this end, given a graph G, we write V(G) its vertex set, and E(G) its edge set. Throughout, we write |G| for |V(G)|, for short. We now specify a notion of vertex boundary that we extensively use.
Define the Cheeger constant (aka bottleneck ratio) of a finite graph G as This quantity is a well-known gauge for the connectivity of a finite set, in particular governing the mixing rates for reversible Markov chains on the set, see [16]. Instead of the vertex boundary in (2.3), more commonly one uses the edge boundary (that is, the set of all edges with one end in A, the other in V(G)\A); in our setting, they differ by at most a factor of (2d − 1), which is unimportant for us. In the sequel, we will be considering (2.3) for G = D n .
In the case of lattice balls B R ⊂ Z d , it is well known that there exists some positive constant c d , such that Φ(B R ) ≥ c d |B R | −1/d for all R > 1, and such isoperimetry is optimal for finite subgraphs of Z d . Given Theorem 2.1, it is pertinent to wonder if the isoperimetry of lattice balls is inherited by IDLA, and the main result of this article is the following.
, and a-priori a cut in D n can be as small as consisting of only one edge. The point here is to show that randomly produced cluster, here IDLA, is much better structured than deterministic ones under the same shape constraints. Nevertheless, we believe that the logarithmic correction in (2.4) should not be there.
For that reason, we do not state the result for d = 2.
Turning to the application to recurrence on {D n } we have in mind, denote by {X n } n∈N a lazy SRW on {D n } with X 0 = 0. Fixing a realization ω of {D n }, its n-th step transition probability is defined to be for some {γ ω n (x)} denoting the probabilities of staying put. In words, each step the random walk either stays put or chooses uniformly among neighbors in its current ECP 24 (2019), paper 31.
cluster. Here, P ω denotes the quenched probability governing the random walk X, and P is reserved for the randomness of {D n }. We will add the superscript ω only when it is helpful to emphasize the randomness of the environment. We say that {X n } is uniformly lazy with probability γ ∈ (0, 1), if γ ω n (x) ≥ γ for all ω, n and x ∈ D ω n . A special case of [11,Proposition 1.13] in our context is the following heat kernel lower bound for {X n } on {D n }, where the uniform laziness is but a technical assumption.
We use here the a-priori weaker definition of recurrence. Definition 2.6. We say that X is recurrent if it has infinite expected occupation time at the origin (hence at every other point), i.e. E

Proof
For large enough n = ω d r d , the IDLA cluster D n concentrates on B r with high probability by Theorem 2.1. We thus denote (3.1) That this will be sufficient for proving Theorem 2.3 is due to a deterministic fact.
Recall that by the optimal isoperimetry satisfied by lattice balls, We separately discuss two cases. (a). If |A 1 | ≥ |A 2 |, then clearly we have that Combining both cases proves the claim.
Our proof relies on partitioning a discrete sphere into subsets of comparable size and diameter. There may be more than one way to achieve this. Here we utilize a notion of almost regular partition of the Euclidean unit sphere S d−1 := {x ∈ R d : x 2 = 1}.  We define the notion of a cone in R d determined by the origin and a closed subset of S d−1 .
Definition 3.5. We say that K = K(T) ⊂ R d is an infinite cone determined by a closed subset T ⊂ S d−1 and the origin, if We then define the notion of a cell which is the intersection of a cone and an annulus in R d . We also need a lemma for a specific hitting probability of (ordinary) SRW on Z d . We defer its proof to the end of the section. Lemma 3.7. Fix d ≥ 3. Let C ⊂ R d be a cell determined by the annulus A(R+δ √ log R, R) and a closed subset T ⊂ S n−1 with max x,y∈T x − y 2 ≤ c p /R, as in Definition 3.6, where R, δ, c p > 1. We further denote C = C ∩ Z d the discrete cell, and Y a SRW on Z d . Then, there exist finite constants R 1 = R 1 (d) and C = C (d, δ, c p ) such that for any R > R 1 , where P v indicates the law of Y when starting from v ∈ Z d .
We now prove the main proposition. √ log r such that x−y 2 ≤ c p for some finite constant c p = c p (d, c ) > 1 and any x, y ∈ T j , j = 1, ..., M . Each partition block T j , j = 1, ..., M and the annulus A(r + a √ log r, r − 3a √ log r) determines a cell C j = C j (A, T j ) in the sense of Definition 3.6. We then denote the discrete cells C j := C j ∩ Z d , j = 1, ..., M .
Let us focus on each fixed cell C j , j ∈ {1, ..., M }. We compute the probability that the m-th particle Y m , m ∈ [n , n], upon being injected from the origin, hits C j before D c m \C j , conditional on S n . To this end, we define the events E (j) m := {Y m hits C j before D c m \C j }, n < m ≤ n, j ∈ {1, .., M }.  By Lemma 3.7, there exists some finite constant C = C (d, a, c p ) such that for any n < m ≤ n and j ∈ {1, ..., M }, Since A ⊂ V(F n ) and F n ∩ D n = ∅, the particle Y k that eventually settles on some x ∈ A must have index n < k = k(x) ≤ n. Further, since ∂ Dn A is a vertex-cutset that separates A from the origin, such Y k must pass through ∂ Dn A at least once before settling on x ∈ A. Let us denote by v = v(x) the first vertex in ∂ Dn A that Y k hits, before settling at x. Since v ∈ V(F n ), it belongs to some cell C j(v) , where j(v) ∈ {1, ..., M }. Consequently, such Y k in order to settle on x ∈ A must hit C j(v) before D c k \C j(v) (since otherwise Y k already settles on the first vertex it encounters in D c k \C j(v) and will not hit v hence will not settle on x).
With A ⊂ V(F n ), ∂ Dn A cannot intersect more than 2|∂ Dn A|∧M number of cells, as the cells are vertex-disjoint except possibly at their boundaries. Further, every x ∈ A is settled by some particle Y k(x) . Thus, the event {∃A ⊂ V(F n ) : |A| ≥ 4C c |∂ Dn A|(log r) 3/2 } implies, by pigeonhole principle, that there exists at least one cell, say C j * , for which the event { Upon taking n large enough such that n ≥ n 0 , by Theorem 2.1 we have that P(∃A ⊂ V(F n ) : |A| ≥ 4C c |∂ Dn A|(log r) 3/2 ) ≤ 2n −2 + c r d−1 e −C c * (log r) 3/2 .
which can be made smaller than 3n −2 upon taking n ≥ n 1 for some large enough n 1 .
Recall that Green's function on Z d , d ≥ 3, is harmonic except at the origin, and satisfies the asymptotics as x 2 → ∞ One also defines Green's function restricted to A ⊂ Z d as the symmetric function It is a consequence of last exit decomposition, see [15,Lemma 6.3.6], that for any x ∈ ∂(K c ), i.e. the outer vertex boundary of K, Uniformly for all z ∈ ∂B R/2 , by [15, Proposition 6.3.5] we have that The first inequality is due to both vertices 0, z ∈ K ⊂ B R+δ √ log R , and the last inequality holds provided we take R > R 1 for some large enough R 1 = R 1 (d). Further, for any x ∈ ∂(K c ), ECP 24 (2019), paper 31. since starting from K c , in order to hit B R/2 , Y must first pass through ∂B R . Since {G(Y k )} k∈N is a martingale provided Y is bounded away from 0, by optional stopping theorem and (3.7), the latter probability is bounded by log R for some dimensional constant c(d). We thus get the desired bound of the form C R 1−d log R for some C = C (d, δ, c p ) finite.

Extensions to continous-time IDLA
We consider simple random walk {X t } t≥0 (2.5) evolving independently on a continuous-time analogue of IDLA {D t } t≥0 on Z d , d ≥ 3, formed by particles injected from the origin at an inhomogeneous Poisson rate λ(t) > 0. As a result, one can achieve quite general growth rate for the cluster by tuning the particles injection rate. Such continuous-time extension of the classical model appears e.g. in [14,10].
More specifically, let {N (t)} t≥0 be an inhomogeneous Poisson process with mean m(t) := t 0 λ(s)ds, satisfying suitable conditions as specified in (4.1). Similarly to the discrete-time case, starting at D 0 = {0}, at each Poisson arrival time, a particle is injected from the origin, performing SRW until settling on the first unoccupied vertex, and then that vertex is added to the cluster. Clearly, by time t, a total of N (t) particles have been injected. We still omit the travel time of the particles and consider them settled instantaneously, therefore |D t | = N (t). Given {D t } t≥0 we perform (lazy) SRW {X t } t≥0 independently on the cluster, jumping at integer times, as specified in (2.5) with n = t .
(This omission of particles travel time is inconsequential when d ≥ 3, as explained already in [14, Section 6] and [10]. That is, our conclusions are unaltered if we allow particles to move simultaneously with each other and with the autonomous random walk X. When d = 2, there is indeed a difference between the two formalisms.) Since both Theorem 2.1 and Theorem 2.3 are properties of fixed large cluster, irrespective of whether the IDLA emission is time-continuous or discrete, we have the following result. Corollary 4.1. Fix d ≥ 3. Assume that there exist some constants 0 ≤ α < d/2 and β > 0, such that the following hold for all t large enough β log t ≤ m(t) ≤ t α . Then, for almost every realization ω of the continuous-time IDLA cluster {D t } t≥0 on Z d , the uniformly lazy SRW {X t } t≥0 evolving independently on {D ω t } t≥0 is recurrent as soon as ∞ 1 |D ω t | −1 dt = ∞. Further, X is recurrent for a.e. realization ω as soon as Therefore, Proposition 2.5 applies to give for some c 0 = c 0 (d, γ) > 0, a.e. ω, and all t sufficiently large .
This immediately implies the results.