Related papers: The Best Mixing Time for Random Walks on Trees
It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on…
We introduce a variation of strong stationary times for random walks on the symmetric group. Rather than proceed in the usual fashion of accumulating larger and larger blocks of cards which may be in any order, we wait for pairs of cards to…
We prove an upper bound for the $\varepsilon$-mixing time of the symmetric exclusion process on any graph G, with any feasible number of particles. Our estimate is proportional to $\mathsf{T}_{\mathsf{RW}(G)}\ln(|V|/\varepsilon)$, where |V|…
We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and…
We study a new optimal stopping problem: Let $G$ be a fixed graph with $n$ vertices which become active on-line in time, one by another, in a random order. The active part of $G$ is the subgraph induced by the active vertices. Find a…
For any given vertices $u$ and $v$ in a graph, the hitting time of a random walk on a finite graph is the number of steps it takes for a random walk to reach vertex $v$ starting at vertex $u$. The expected value of the hitting time is the…
We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of the $n$-vertex complete $b$-ary tree when $3\leq q\leq b/2\ln(b)$. We give both upper and lower bounds on the mixing time. For fixed $q$ and $b$, our upper bound…
The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion…
We provide the first algorithm for computing an optimal tree decomposition for a given graph $G$ that runs in single exponential time in the feedback vertex number of $G$, that is, in time $2^{O(\text{fvn}(G))}\cdot n^{O(1)}$, where…
We introduce the concept of Most, and Least, Compact Spanning Trees - denoted respectively by $T^*(G)$ and $T^\#(G)$ - of a simple, connected, undirected and unweighted graph $G(V, E, W)$. For a spanning tree $T(G) \in \mathcal{T}(G)$ to be…
We consider a random geometric graph obtained by placing a Poisson point process of intensity 1 in the d-dimensional torus of side length n^(1/d) and connecting two points by an edge if their distance is at most r. We consider the case of…
This paper considers non-backtracking random walks on random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph…
We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches…
We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set…
Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows.…
A random walk on a regular tree (or any non-amenable graph) has positive speed. We ask whether such a walk can be slowed down by applying carefully chosen time-dependent permutations of the vertices. We prove that on trees the random walk…
We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite…
We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine…
We study a natural random walk on the $n \times n$ upper triangular matrices, with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this…
We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.