Related papers: The Best Mixing Time for Random Walks on Trees

We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a…

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $O(1/\epsilon^2)$, where $\epsilon$ is the step…

Random walks have wide application in real lives, ranging from target search, reaction kinetics, polymer chains, to the forecast of the arrive time of extreme events, diseases or opinions. In this paper, we consider discrete random walks on…

For a connected graph $G$, the average hitting time $\alpha(G)$ and the Kemeny's constant $\kappa(G)$ are two similar quantities, both measuring the time for the random walk on $G$ to travel between two randomly chosen vertices. We prove…

We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive `step' or a multiplicative `jump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the…

We show bounds on total variation and $L^{\infty}$ mixing times, spectral gap and magnitudes of the complex valued eigenvalues of a general (non-reversible non-lazy) Markov chain with a minor expansion property. This leads to the first…

We study a random walk on $\mathbb{F}_p$ defined by $X_{n+1}=1/X_n+\varepsilon_{n+1}$ if $X_n\neq 0$, and $X_{n+1}=\varepsilon_{n+1}$ if $X_n=0$, where $\varepsilon_{n+1}$ are independent and identically distributed. This can be seen as a…

We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree…

It is a classic result in spectral theory that the limit distribution of the spectral measure of random graphs G(n, p) converges to the semicircle law in case np tends to infinity with n. The spectral measure for random graphs G(n, c/n)…

We prove a cutoff for the random walk on random $n$-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$ is a…

A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution…

We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…

In this paper, we present a flexible and probabilistic framework for tracking topological features in time-varying scalar fields using merge trees and partial optimal transport. Merge trees are topological descriptors that record the…

The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as \emph{mixing}, finds widespread applications in several areas of quantum information and computation. Of particular interest in most…

We investigate the statistics of extremal path(s) (both the shortest and the longest) from the root to the bottom of a Cayley tree. The lengths of the edges are assumed to be independent identically distributed random variables drawn from a…

For a graph $G$, an edge-separating (resp. vertex-separating) path system of $G$ is a family of paths in $G$ such that for any pair of edges $e_1, e_2$ (resp. pair of vertices $v_1, v_2$) of $G$ there is at least one path in the family that…

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 of time randomly rewiring a fraction…

We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of…

We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $\mu \ll 1/n$, and is then declared open with probability $p = \lambda/n$ where $\lambda > 1$. We study a random walk on this…

Node connectivity plays a central role in temporal network analysis. We provide a comprehensive study of various concepts of walks in temporal graphs, that is, graphs with fixed vertex sets but edge sets changing over time. Taking into…