Reinforced Random Walks
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.
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1.11
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.
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.
11
1.11
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.
References: 17 7 [62]] A. M. Vershik and S. V. Kerov. Asymptotics of the Plancherel measure of the symmetricc group and the limiting form of Young tables. Soviet Math. DokL, 18:527531,, 1977. [63]] M. Vervoort. Games, Walks and Grammars. Problems I 've worked on. PhD thesis, Universiteitt van Amsterdam, 2000. [64]] S. Volkov. Vertex-reinforced random walk on arbitrary graphs. 29(1):66-91,, 2001. Ann. Probab., [65]] S. L. Zabell. W. E. Johnson 's "sufficientness" postulate. Ann. Statist, 10(4):109010999 (1 plate), 1982. [66]] M. P. W. Zerner and F. Merkl. A zero-one law for planar random walks in random environment.. Preprint, 2000. 188 Chapter 1. Introduction