Related papers: Aldous-type spectral gap results for the complete …

We study random walks on the giant component of the Erd\H{o}s-R\'enyi random graph ${\cal G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is a non-compact simple Lie group and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup.…

Suppose we are given finitely generated groups $\Gamma_1,...,\Gamma_m$ equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic…

A symmetric branching random walk (BRW) on a free group $\mathbb{F}$ is transient if and only if the mean offspring number $r$ does not exceed $R$, the reciprocal of the spectral radius of the underlying random walk. In this regime, the…

We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers…

We study the mixing time of a non-Markovian process, the step-reinforced random walk (SRRW) on a finite group. This process differs from a classical random walk in that at each integer time, with probability $\alpha$ the next step is chosen…

Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work…

In this paper we introduce a new notion of a sequence of symmetry groups of an infinite word. Given a subgroup $G_n$ of the symmetric group $S_n$, it acts on the set of finite words of length $n$ by permutation. We associate to an infinite…

We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ram\'irez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists of a particular version of the model studied in [arXiv…

In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is…

Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The results of this article supplement those in the three main papers on random Cayley…

We say that a subset $S\subseteq F_N$ is \emph{spectrally rigid} if whenever $T_1, T_2\in cv_N$ are points of the (unprojectivized) Outer space such that $||g||_{T_1}=||g||_{T_2}$ for every $g\in S$ then $T_1=T_2$ in $\cvn$. It is…

In this paper, we study random walks $g_n=f_{n-1}\cdots f_0$ on the group $\mathrm{Homeo}(S^1)$ of the homeomorphisms of the circle, where the homeomorphisms $f_k$ are chosen randomly, independently, with respect to a same probability…

We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle $\mathbb{Z}/n\mathbb{Z}$. One starts with a mass density $\mu>0$ of initially active particles, each of which performs a simple symmetric random…

In this paper, we study the spectral gap and principle eigenfunction of the random walk in the line segment $[1, N]$ with conductances $c^{(N)}(x, x+1)_{1\le x<N}$ where $c^{(N)}(x, x+1)>0$ is the rate of the random walk jumping from site…

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i)…

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder chain on a graph $G=(V,E)$ is a discrete-time stochastic process with values in the space of rooted trees…

A Cayley graph on the symmetric group $S_n$ is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of…

We consider a model of a random height function with long-range constraints on a discrete segment. This model was suggested by Benjamini, Yadin and Yehudayoff and is a generalization of simple random walk. The random function is uniformly…

Let $(X_n)_{n\ge 1}$ be a Markov chain on a measurable state space $X$, and let $S_n = \sum_{k=1}^n f(X_k)$ be the associated Markov walk. For $y>0$, denote by $\tau_y$ the first time at which $y+S_n$ becomes non-positive. Assuming that the…