Cutoff for non-backtracking random walks on sparse random graphs

@article{BenHamou2015CutoffFN,
  title={Cutoff for non-backtracking random walks on sparse random graphs},
  author={Anna Ben-Hamou and Justin Salez},
  journal={arXiv: Probability},
  year={2015}
}
A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non… 

Figures from this paper

Random walk on sparse random digraphs
A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter
Comparing mixing times on sparse random graphs
TLDR
The precise worst-case mixing time for simple random walk on G is determined, and it is shown that this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton-Watson tree.
Cutoff at the “entropic time” for sparse Markov chains
We study convergence to equilibrium for a class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix P the mass is essentially
A threshold for cutoff in two-community random graphs
In this paper, we are interested in the impact of communities on the mixing behavior of the non-backtracking random walk. We consider sequences of sparse random graphs of size $N$ generated according
CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS
It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a
Limit profiles for reversible Markov chains
In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the
Random walks on dynamic configuration models:A trichotomy
We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit
Mixing times of random walks on dynamic configuration models
The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we
Concentration and compression over infinite alphabets, mixing times of random walks on random graphs
This document presents the problems I have been interested in during my PhD thesis. I begin with a concise presentation of the main results, followed by three relatively independent parts. In the
Mixing time of PageRank surfers on sparse random digraphs
TLDR
This trichotomy is shown to hold uniformly in the starting point and for a large class of distributions $\lambda$, including widespread as well as strongly localized measures.
...
1
2
...

References

SHOWING 1-10 OF 36 REFERENCES
Non-backtracking random walks mix faster
We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as
Cutoff phenomena for random walks on random regular graphs
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
Characterization of cutoff for reversible Markov chains
A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a
Random Walks on Finite Groups
TLDR
This article gives a general overview of Markov chains on finite sets, and how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups.
Cutoff for conjugacy-invariant random walks on the permutation group
We prove a conjecture raised by the work of Diaconis and Shahshahani (Z Wahrscheinlichkeitstheorie Verwandte Geb 57(2):159–179, 1981) about the mixing time of random walks on the permutation group
The cutoff phenomenon in finite Markov chains.
  • P. Diaconis
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 1996
TLDR
This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn), and shows that chains with polynomial growth (drunkard's walk) do not show cutoffs.
Tight inequalities among set hitting times in Markov chains
Given an irreducible discrete time Markov chain on a finite state space, we consider the largest expected hitting time T(α) of a set of stationary measure at least α for α ∈ (0, 1). We obtain tight
On the second eigenvalue of random regular graphs
  • A. Broder, E. Shamir
  • Computer Science, Mathematics
    28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
  • 1987
TLDR
It is shown that the second eigenvalue of d-regular graphs, λ2, is concentrated in an interval of width O(√d) around its mean, and that its mean is O(d3/4).
Mixing and hitting times for finite Markov chains
Let $0<\alpha<1/2$. We show that that the mixing time of a continuous-time Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of the state space
Universality of cutoff for the ising model
On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s.
...
1
2
3
4
...