Random walk on barely supercritical branching random walk

Abstract

Let T be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean µ > 1, conditioned to survive. Let ϕ T be a random embedding of T into Z d according to a simple random walk step distribution. Let T p be percolation on T with parameter p, and let p c = µ −1 be the critical percolation parameter. We consider a random walk (X n ) n≥1 on T p and investigate the behavior of the embedded process ϕ Tp (X n ) as n → ∞ and simultaneously, T p becomes critical, that is, p = p n p c . We show that when we scale time by n/(p n − p c ) 3 and space by (p n − p c )/n, the process (ϕ Tp (X n )) n≥1 converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.MSC 2010. Primary: 60K37, 82C41. Secondary: 60F17, 60K40. Key words and phrases. Branching random walk, random walk indexed by a tree, percolation, scaling limit, supercriticality. arXiv:1804.04396v2 [math.PR] 13 Mar 2019 1.1. Random walk on a randomly embedded random tree. Before we proceed with the main results, let us define the model.Percolation on Galton-Watson trees. Let T be a Galton-Watson tree rooted at with law P, and let ξ be the random number of offspring of the root. Suppose that the tree is supercritical, i.e., µ := E[ξ] > 1. We write ∆ for the maximal number of children that a single individual in the (unpercolated) tree can have, i.e., ∆ := sup{n : P(ξ = n) > 0}.We consider percolation on the edges of the tree and let T p denote the connected component of the root in T , when each edge is deleted with probability 1 − p ∈ (0, 1), independently of every other edge and of T . Then T p is again a Galton-Watson tree, whose distribution we denote by P p . The root of a tree T chosen according to P p now has ξ p offspring, such that, conditioned on {ξ = n}, ξ p is a binomial random variable with parameters n and p.The percolated tree has mean number of offspring pµ and is supercritical if and only if p > p c := 1/µ. In this setting, denote byP p = P p ( · | |T | = ∞) the distribution of the percolated tree, conditioned on non-extinction. Given a tree T rooted at , let (X n ) n≥0 be a simple random walk on T started at . Given v ∈ T , we write P v T for the law of (X n ) n≥0 with X 0 = v, and write P T if v = . Given a realization of a Galton-Watson tree T , we call P T the quenched law of the random walk, and we call P p :=P p × P T the annealed law of the random walk on the tree (writing P v p :Spatial embedding: branching random walk. We embed a Galton-Watson tree T into Z d by means of a branching random walk, which we will define now: Let D denote a non-degenerate probability distribution on Z d . Given a tree T rooted at , set Z( ) = 0 and assign to each vertex v = of T an independent random variable Z(v) with law D. For any v ∈ T there exists a unique path = v 0 , v 1 , . . . , v m = v. The branching random walk embedding ϕ = ϕ T is defined such that ϕ(v) :=...

The speed of random walk on Galton-Watson trees with vanishing conductances

“…Closely related are results for random walks in Z d with random conductances and bias parameter λ, as studied by [17,18,7], or for the Mott random walk [15,16]. The regularity of the speed on the tree as a function of the offspring law was studied in [20], when the offspring law is close to criticality.…”

The speed of random walk on Galton-Watson trees with vanishing conductances

“…Closely related are results for random walks in Z d with random conducances and bias parameter λ, as studied by [FH14,GGN17,BGN19], or for the Mott random walk [FGS18,FS19]. The regularity of the speed on the tree as a function of the offspring law was studied in [vdHHN20], when the offspring law is close to criticality.…”

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