Vertex-reinforced Random Walk on Z with Sub-square-root Weights Is Recurrent
نویسندگان
چکیده
We prove that vertex-reinforced random walk on Z with weight of order kα, for α ∈ [0, 1/2), is recurrent. This confirms a conjecture of Volkov for α < 1/2. The conjecture for α ∈ [1/2, 1) remains open.
منابع مشابه
Zeno's Walk: a Random Walk with Refinements
A self-modifying random walk on Q is deened from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which is supported on a subset of R having Hausdorr dimension less than 1. By generating function techniques we then calcul...
متن کاملExcited Random Walk in Three Dimensions Has Positive Speed
Excited random walk is a model of a random walk on Z which, whenever it encounters a new vertex it receives a push toward a specific direction, call it the “right”, while when it reaches a vertex it “already knows”, it performs a simple random walk. This model has been suggested in [BW] and had since got lots of attention, see [V, Z]. The reason for the interest is that it is situated very natu...
متن کاملVertex - 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 ...
متن کاملLevel Crossing Probabilities Ii: Polygonal Recurrence of Multidimensional Random Walks
In part I (math.PR/0406392) we proved for an arbitrary onedimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n). In higher dimensions we call a random walk ’polygonally recurrent’ (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above e...
متن کاملDirectionally Reinforced Random Walks
This paper introduces and analyzes a class of directionally reinforced random walks The work is motivated by an elementary model for time and space correlations in ocean surface wave elds We develop some basic properties of these walks For instance we investigate recurrence properties and give conditions under which the limiting continuous versions of the walks are Gaussian di usion processes M...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014