Biased random walks on random graphs

These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our main focus will be on directionally transient and reversible random walks on different types of underlying graph structures, such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.Comment: Survey based one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. 64 pages, 16 figure

Random walks on quasirandom graphs

Let G be a quasirandom graph on n vertices, and let W be a random walk on G of length alpha n^2. Must the set of edges traversed by W form a quasirandom graph? This question was asked by B\"ottcher, Hladk\'y, Piguet and Taraz. Our aim in this paper is to give a positive answer to this question. We also prove a similar result for random embeddings of trees.Comment: 19 pages, 2 figure

On the trace of random walks on random graphs

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random graph $G\sim G(n,p)$ for $p>C\ln{n}/n$ is, with high probability, Hamiltonian and $\Theta(\ln{n})$-connected. In the special case $p=1$ (i.e. when $G=K_n$), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the $k$'th time, the trace becomes $2k$-connected.Comment: 32 pages, revised versio

High Dimensional Random Walks and Colorful Expansion

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 we {\em define high order random walks}: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension $d$ has vertices, edges, triangles, pyramids, up to $d$-dimensional cells. For any $0 \leq i < d$, a high order random walk on dimension $i$ moves between neighboring $i$-faces (e.g., edges) of the complex, where two $i$-faces are considered neighbors if they share a common $(i+1)$-face (e.g., a triangle). The case of $i=0$ recovers the well studied random walk on graphs. We provide a {\em local-to-global criterion} on a complex which implies {\em rapid convergence of all high order random walks} on it. Specifically, we prove that if the $1$-dimensional skeletons of all the links of a complex are spectral expanders, then for {\em all} $0 \le i < d$ the high order random walk on dimension $i$ converges rapidly to its stationary distribution. We derive our result through a new notion of high dimensional combinatorial expansion of complexes which we term {\em colorful expansion}. This notion is a natural generalization of combinatorial expansion of graphs and is strongly related to the convergence rate of the high order random walks. We further show an explicit family of {\em bounded degree} complexes which satisfy this criterion. Specifically, we show that Ramanujan complexes meet this criterion, and thus form an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.Comment: 27 page

On the physical relevance of random walks: an example of random walks on a randomly oriented lattice

Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent results on random walks on randomly oriented lattices. It is worth noticing that general undirected graphs are associated with (not necessarily Abelian) groups while directed graphs are associated with (not necessarily Abelian) $C^*$-algebras. Since quantum mechanics is naturally formulated in terms of $C^*$-algebras, the study of random walks on directed lattices has been motivated lately by the development of the new field of quantum information and communication

Viral processes by random walks on random regular graphs

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph, and $\epsilon$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $\beta\in(0,1)$ determined by the parameters of the model, $\beta k$ get infected with probability $\beta$, and $O(\ln k)$ get infected with probability $(1-\beta)$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.Comment: Published in at http://dx.doi.org/10.1214/13-AAP1000 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org