相关论文: Vertex-Reinforced Random Walk

In this paper, we propose and analyze a novel one-dimensional inhomogeneous random walk model that combines spatial decay of transition probabilities with a temporal renewal structure for each excursion. In this model, the probability of…

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…

We consider the problem of stochastic flow of multiple particles traveling on a closed loop, with a constraint that particles move without passing. We use a Markov chain description that reduces the problem to a generalized random walk on a…

We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and…

We consider finite-state time-nonhomogeneous Markov chains where the probability of moving from state $i$ to state $j\neq i$ at time $n$ is $G(i,j)/n^\zeta$ for a ``generator'' matrix $G$ and strength parameter $\zeta>0$. In these chains,…

We solve an adaptive search model where a random walker or L\'evy flight stochastically resets to previously visited sites on a $d$-dimensional lattice containing one trapping site. Due to reinforcement, a phase transition occurs when the…

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks in the quarter plane are characterized by the fact that the one-step transition probabilities…

We consider a system of interacting random walks known as the frog model. Let $\mathcal{K}_n=(\mathcal{V}_n,\mathcal{E}_n)$ be the complete graph with $n$ vertices and $o\in\mathcal{V}_n$ be a special vertex called the root. Initially,…

By a theorem of Volkov (2001) we know that on most graphs with positive probability the linearly vertex-reinforced random walk (VRRW) stays within a finite "trapping" subgraph at all large times. The question of whether this tail behavior…

Diffusion in a confining potential offers a minimal setting to understand the interplay between random motion and deterministic forces driving a particle towards a focal point or potential minimum. In continuous space and time, two…

We describe and analyze how reinforced random walks can eventually localize, i.e. only visit finitely many sites. After introducing vertex and edge self-interacting walks on a discrete graph in a general setting, and stating the main…

We introduce the concept of a deterministic walk. Confining our attention to the finite state case, we establish hypotheses that ensure that the deterministic walk is transitive, and show that this property is in some sense robust. We also…

We consider the statistics of occupation times, the number of visits at the origin and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these…

We consider an elementary model for self-organised criticality, the activated random walk on the complete graph. We introduce a discrete time Markov chain as follows. At each time step, we add an active particle at a random vertex and let…

We study the transition probability, say $p_A^n(x,y)$, of a one-dimensional random walk on the integer lattice killed when entering into a non-empty finite set $A$. The random walk is assumed to be irreducible and have zero mean and a…

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit…

Consider a critical nearest neighbor branching random walk on the $d$-dimensional integer lattice initiated by a single particle at the origin. Let $G_{n}$ be the event that the branching random walk survives to generation $n$. We obtain…

Deterministic walk in an excited random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of…

Let $X$ be a random walk on the torus of side length $N$ in dimension $d\geq 3$ with uniform starting point, and $t_{\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the…

We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and…