Related papers: Local $\epsilon$-uniform mixing in continuous quan…

We show sufficient conditions under which the \textsc{BallWalk} algorithm mixes fast in a bounded connected subset of $\Real^n$. In particular, we show fast mixing if the space is the transformation of a convex space under a smooth…

An $\epsilon$-distance-uniform graph is one in which from every vertex, all but an $\epsilon$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in…

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 quantum particle evolving by Schr\"odinger's equation in discrete space constitutes a continuous-time quantum walk on a graph of vertices and edges. When a vertex is marked by an oracle, the quantum walk effects a quantum search…

We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state…

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of…

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples…

Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral…

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 investigate what happens when the random…

We define the Uniform Random Walk (URW) on a connected, locally finite graph as the weak limit of the uniform walk of length $n$ starting at a fixed vertex. When the limit exists, it is necessarily Markovian and is independent of the…

In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a…

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…

In this paper, we are interested in the mixing behaviour of simple random walks on inhomogeneous directed graphs. We focus our study on the Chung-Lu digraph, which is an inhomogeneous network that generalizes the Erd\H{o}s-R\'enyi digraph.…

We study the continuous-time quantum walks on graphs in the adjacency algebra of the $n$-cube and its related distance regular graphs. For $k\geq 2$, we find graphs in the adjacency algebra of $(2^{k+2}-8)$-cube that admit instantaneous…

If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights…

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $n^{1/4 + o_n(1)}$ in $n$ units of time. Together with the complementary lower bound proven by Gwynne and…

Consider an ergodic unimodular random one-ended planar graph $\G$ of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By "locally…

The lackadaisical quantum walk is a discrete-time, coined quantum walk on a graph with a weighted self-loop at each vertex. It uses a generalized Grover coin and the flip-flop shift, which makes it equivalent to Szegedy's quantum Markov…

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…

Due to the unitary evolution, quantum walks display different dynamical features from that of classical random walks. In contrast to this expectation, in this work, we show that extreme events can arise in unitary dynamics and its…