Vertex-reinforced random walk on ℤ eventually gets stuck on five points

Vertex - Reinforced Random Walk on Z Eventually Gets Stuck on Five Points

Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the one-dimensional integer lattice Z. They proved that the range is almost surely finite and that with positive probability the ...

Vertex-reinforced random walk on Z has finite range

A stochastic process called Vertex-Reinforced Random Walk (VRRW) is defined in Pemantle (1988a). We consider this process in the case where the underlying graph is an infinite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely finite, that at least 5 points are visited infinitely often almost surely, and that with positive probability the range contains e...

Vertex-reinforced Random Walk on Z Has Nite Range

A stochastic process called Vertex-Reinforced Random Walk (VRRW) is deened in Pe-mantle (1988a). We consider this process in the case where the underlying graph is an innnite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely nite, that at least 5 points are visited innnitely often almost surely, and that with positive probability the range contains exact...

Vertex-Reinforced Random Walk on Arbitrary Graphs

Vertex-reinforced Random Walk

This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1, . . . , d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix, R, which is real, symmetric and nonnegative. Let Si(n) keep track of the number of visits to state i up to time n, and form the f...