Limiting behavior for the distance of a random walk

Berestycki, Nathanaël; Durrett, Rick

doi:10.1214/ejp.v13-490

Cited by 10 publications

(16 citation statements)

References 23 publications

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“…It is well-known (see [8,13,18,28]) thatD, the average distance between two vertices in C 1 , is log λ n + O p (1), analogous to the fact thatD = log d−1 n + O p (1) in G(n, d), the uniform d-regular graph on n vertices 1 (both are locally-tree-like: G(n, d) resembles a d-regular tree while G(n, p) resembles a Poisson(λ)-GW tree). It is then natural to expect that t (v 1 ) mix coincides with the time it takes the walk to reach this typical distance from its origin v 1 , which would be ν −1D for a random walk on a Poisson(λ)-GW tree Supporting evidence for this on the random 3-regular graph G(n, 3) was given in [6], where it was shown that the distance of the walk from v 1 after t = c log n steps is w.h.p. (1 + o(1))(νt ∧D), with ν = 1 3 being the speed of random walk on a binary tree.…”

Section: Introductionmentioning

confidence: 94%

Random walks on the random graph

Berestycki¹,

Lubetzky²,

Peres³

et al. 2018

Ann. Probab.

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We study random walks on the giant component of the Erdős-Rényi random graph G(n, p) where p = λ/n for λ > 1 fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order log 2 n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(log n) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (νd) −1 log n ± (log n) 1/2+o(1) , where ν and d are the speed of random walk and dimension of harmonic measure on a Poisson(λ)-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.

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Section: Introductionmentioning

confidence: 94%

Random walks on the random graph

Berestycki¹,

Lubetzky²,

Peres³

et al. 2018

Ann. Probab.

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“…A special case of this was conjectured by Durrett, following his work with Berestycki [8] studying the SRW on a random 3-regular graph G ∼ G(n, 3). They showed that at time c log 2 n the distance of the walk from its starting point is asymptotically ( c 3 ∧ 1) log 2 n. This implies a lower bound of 3 log 2 n for the asymptotic mixing time of random 3-regular graphs, and in particular, an asymptotic lower bound of 6 log 2 n for the lazy random walk (the lazy version of a chain with transition kernel P is the chain whose transition kernel is 1 2 (P + I), i.e., in each step it stays in place with probability 1 2 , and otherwise it follows the rule of the original chain).…”

Section: Introductionmentioning

confidence: 96%

Cutoff phenomena for random walks on random regular graphs

Lubetzky

Sly

2010

Duke Math. J.

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The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on $\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that almost every such graph for $d\geq 3$ is an expander, and even essentially Ramanujan, implying a mixing-time of $O(\log n)$. According to a conjecture of Peres, the simple random walk on $\G(n,d)$ for such $d$ should then exhibit cutoff with high probability. As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is w.h.p. $(6+o(1))\log_2 n$. In this work we confirm the above conjectures, and establish cutoff in total-variation, its location and its optimal window, both for simple and for non-backtracking random walks on $\G(n,d)$. Namely, for any fixed $d\geq3$, the simple random walk on $\G(n,d)$ w.h.p. has cutoff at $\frac{d}{d-2}\log_{d-1} n$ with window order $\sqrt{\log n}$. Surprisingly, the non-backtracking random walk on $\G(n,d)$ w.h.p. has cutoff already at $\log_{d-1} n$ with constant window order. We further extend these results to $\G(n,d)$ for any $d=n^{o(1)}$ that grows with $n$ (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.Comment: 33 pages, 4 figure

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“…In order to explain this, we show that the equilibrium state can be viewed as the stationary measure of an effective split-merge process. This strategy was recently applied successfully by Schramm to the random interchange model on the complete graph [22] (the result was first conjectured by Aldous, see [6]). The absence of spatial structure makes the situation much simpler, but it was nonetheless a tour de force to prove that long cycles occur, that they satisfy an effective split-merge process, and that their asymptotic distribution is Poisson-Dirichlet (see also [5] for subsequent simplifications and improvements).…”

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confidence: 99%

Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

2012

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We study random spatial permutations on Z 3 where each jump x → π(x) is penalized by a factor e −T x−π(x) 2 . The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.

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Limiting behavior for the distance of a random walk

Cited by 10 publications

References 23 publications

Random walks on the random graph

Random walks on the random graph

Cutoff phenomena for random walks on random regular graphs

Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

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