Related papers: Restrictions of some reinforced processes to subgr…
We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function $w$ is reciprocally integrable and non-decreasing, then…
In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarr\`es of the…
We introduce random interlacements for transient vertex-reinforced jump processes on a general graph $G$. Using increasing finite subgraphs $G_n$ of $G$ with wired boundary conditions, we show convergence of the vertex-reinforced jump…
In this paper we introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of…
We prove that the only nearest neighbor jump process with local dependence on the occupation times satisfying the partial exchangeability property is the vertex reinforced jump process, under some technical conditions. This result gives a…
This article introduces a model for interacting vertex-reinforced random walks, each taking values on a complete sub-graph of a locally finite undirected graph. The transition probability for a walk to a given vertex depends on the…
Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process, which takes values in the vertex set of a graph $G$, and is more likely to cross edges it has visited before. We show that it can be…
The aim of our work is to study vertex-reinforced jump processes with super-linear weight function $w(t) = t^\alpha$ , for some $\alpha>1$. On any complete graph $G = (V, E)$, we prove that there is one vertex $v \in V$ such that the total…
We give a simple proof for recurrence of vertex reinforced jump process on \(\mathbb{Z}^d\), under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on \ \(\mathbb{Z}^d\) is…
Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \ge 3$. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the…
This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. We show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a…
We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the…
We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently…
Graph vertex embeddings based on random walks have become increasingly influential in recent years, showing good performance in several tasks as they efficiently transform a graph into a more computationally digestible format while…
We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on…
We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us…
We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from $\mathbb{Z}^2$ by replacing every edge by a sufficiently large, but fixed number of edges in…
Vertex-reinforced random walk is defined in Pemantle's (1988) thesis; it is a random walk that is biased to visit sites it has already visited a lot. We show that this reinforcement scheme, in contrast to the scheme of edge-reinforcement,…
A restricted permutation of a locally finite directed graph $G=(V,E)$ is a vertex permutation $\pi: V\to V$ for which $(v,\pi(v))\in E$, for any vertex $v\in V$. The set of such permutations, denoted by $\Omega(G)$, with a group action…
We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\in G}$, the…