Reinforced Random Walks

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Chapterr 1 Introduction n
Thiss thesis contains work on reinforced random walks, the reconstruction of random sceneriess observed along a random walk path, and the length of a longest increasing subsequencee in a random permutation. In this introduction, I will survey some of the work inn the area and describe my results. Furthermore I will explain how all three subjects fit intoo the framework of random walks in stochastic surroundings. Section 1 is dedicated to reinforcedd random walks. Section 2 describes scenery reconstruction problems. Section 33 deals with random permutations and explains the connection with up-right paths in a Poissoniann field.


Reinforced r a n d o m walks
A short history

Differentt surveys on random processes with reinforcement have been written by Pemantle [47],, Davis [7], and Benaïm [4]. My emphasis here is on work which has connections with myy own results. Randomm walks with edge reinforcement Reinforcedd random walks were invented by Coppersmith and Diaconis in 1987 (see [11]). Theyy introduced edge-reinforced random walk, a nearest-neighbor random walk on a locallyy finite graph, as follows: All edges are given strictly positive numbers as weights. In eachh step, the random walker jumps to a nearest-neighbor vertex traversing an edge e incidentt to her current location with probability proportional to the weight of e. Each timee an edge is traversed, its weight is increased by 1. The process remembers where it hass been before and prefers edges which have been traversed often in the past. Edgereinforcedd random walk can be considered as a simple model for a person exploring a neww city. First she traverses randomly the streets around her hotel. As a street becomes familiarr to her, she has a higher preference to traverse the street again in the future. Inn a special case, edge-reinforced random walk is well-known. Consider the graph whichh consists of two vertices w, v and two parallel edges e, ƒ connecting them. The sequencee of edges traversed by edge-reinforced random walk on this graph is a Polya urn process.. Recall that in a Polya urn process balls are drawn from an urn containing balls withh label e and ƒ; after each drawing the ball is returned with an additional ball with the 33


ChapterChapter 1. Introduction

samee label. The process was introduced by Eggenberger and Polya [14] in 1923. We will seee below that some well-known properties for the Polya urn process can be generalized forr edge-reinforced random walk on a general finite graph. Coppersmithh and Diaconis (see [11]) were interested in the asymptotic behavior of the locall time for a finite graph. Generalizations of their model were introduced by Davis [6],, among others reinforced random walks oj sequence type. In this model, a sequence 66kk > 0, k > 1 is given, and the weight of an edge is increased by 6k after the k traversal.. If all edges have initially weight 1, the weight of an edge after k traversals equalss Dk :— l + ^f=i & Tóth [59] considered weakly reinforced random walk] he assumed DDkk ~ kp for k — oo and p e]0,1[. In one dimension, he proved limit theorems for the locall time process and the position of the random walker at late times. Recently Limic [28]] showed that for strong reinforcement. (Dk — kp with p > 1) the process will become "stuck"" eventually, traversing the same edge back and forth. Recurrence e Althoughh people have been studying reinforced random walks for almost 15 years, many fundamentall questions remain open, for instance the recurrence question. We call a randomm walk path recurrent if it visits all vertices infinitely often and transient if it visits alll vertices at most finitely often. We call a random walk recurrent (transient) if almost all itss paths are recurrent (transient). For simple random walk Polya [50] proved recurrence onn Z and Z 2 and transience on Z d for d > 3. Kakutani is reported to have said: "A drunkenn man will eventually find his...
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