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

Traversals of Infinite Graphs with Random Local Orientations

We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks. We define analogues in the setting of random basic walks of the notions of recurrence and transience in the theory of simple random walks, and we study the question of which graphs have a cycling random basic walk and which a transient random basic walk. We prove that cycles of arbitrary length are possible in any regular graph, but that they are unlikely. We give upper bounds on the expected number of vertices a random basic walk will visit on the infinite graphs studied and on their finite analogues of sufficiently large size. We then study random basic walks on complete graphs, and prove that the class of complete graphs has random basic walks asymptotically visit a constant fraction of the nodes. We end with numerous conjectures and problems for future study, as well as ideas for how to approach these problems.Comment: This is my masters thesis from Wesleyan University. Currently my advisor and I are selecting a journal where we will submit a shorter version. We plan to split this work into two papers: one for the case of infinite graphs and one for the finite case (which is not fully treated here

On the trace of branching random walks

We study branching random walks on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for the simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any (non-trivial) branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All results are stated in the more general context of branching random walks on unimodular random graphs.Comment: revised versio

Comparing mixing times on sparse random graphs

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show that this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton-Watson tree