Related papers: Random Walks on Graphs and Approximation of L2-Inv…

Reinforced random walks are random walks on graphs whose transition probabilities along edges from a vertex are proportional to the weights of those edges, but where the weight of an edge evolves in a way that depends on the past traversals…

We initiate the study of what we refer to as random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that the labeling occurs increasingly whenever an unlabeled vertex is…

If $\mathcal{W}$ is the simple random walk on the square lattice $\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing $\mathcal{W}$ as a…

We consider random walks and L\'evy processes in a homogeneous group $G$. For all $p > 0$, we completely characterise (almost) all $G$-valued L\'evy processes whose sample paths have finite $p$-variation, and give sufficient conditions…

Consider $G=SL_2(\mathbb{Z})/\{\pm I\}$ acting on the complex upper half plane $H$ by $h_M(z)=\frac{az+b}{cz+d},$ for $M \in G$. Let $D=\{z \in H: |z|\geq 1, |\Re(z)|\leq 1/2\}$. We consider the set $\mathcal{E} \subset G$ with the $9$…

The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function $W$ taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting…

We review various features of the statistics of random paths on graphs. The relationship between path statistics and Quantum Mechanics (QM) leads to two canonical ways of defining random walk on a graph, which have different statistics and…

Consider $n$ points independently sampled from a density $p$ of class $\mathcal{C}^2$ on a smooth compact $d$-dimensional sub-manifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the generator of a random walk visiting these points…

Large unweighted directed graphs are commonly used to capture relations between entities. A fundamental problem in the analysis of such networks is to properly define the similarity or dissimilarity between any two vertices. Despite the…

Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $|\cdot |$. We study the behavior of the probability of return for random walks driven by symmetric measures $\mu$…

Let $G$ be a finite group and let $H$ be a subgroup of $G$. The left-invariant random walk driven by a probability measure $w$ on $G$ is the Markov chain in which from any state $x \in G$, the probability of stepping to $xg \in G$ is…

We study the escape probability problem in random walks over graphs. Given vertices, $s,t,$ and $p$, the problem asks for the probability that a random walk starting at $s$ will hit $t$ before hitting $p$. Such probabilities can be…

Random walks over directed graphs are used to model activities in many domains, such as social networks, influence propagation, and Bayesian graphical models. They are often used to compute the importance or centrality of individual nodes…

The Rado graph, also known as the random graph $G(\infty, p)$, is a classical limit object for finite graphs. We study natural ball walks as a way of understanding the geometry of this graph. For the walk started at $i$, we show that order…

We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a…

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present paper, we develop a spectral analysis for lamplighter random…

Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to…

We revisit a simple model class for machine learning on graphs, where a random walk on a graph produces a machine-readable record, and this record is processed by a deep neural network to directly make vertex-level or graph-level…

Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the…

Motivated by a recent work of Benoist and Quint and extending results from the PhD thesis of the third author, we obtain limit theorems for products of independent and identically distributed elements of GLd (R), such as the…