Self-attracting random walks

Abstract

We discuss in this chapter a number of problems of random walks with self- attracing path ineteractions which are all closely related to large deviation theory. A simple case of an attraction would be just to change sign in the (weakly) self repellent case of Chapter 1. For technical reasons, it is convenient to work with continuous time but discrete state space Markov processes. Therefore, we consider the standard symmetric random walk on ℤ _d starting in 0 having holding times with expectation 1/d. The path measure on the space D ^∞ = D([0,∞), ℤ _d for the set of paths of length T. As usual, we write X ^t (ω)= ω^t, ω∈ D ∞ for the evaluation map. We then transform the path measure in the same way as in the weakly repellent case, just having the opposite sign of the coupling constant:

$$ \mathbb{Z}^d $$

starting in 0 having holding times with expectation 1/{itd}. The path measure on the space

$$ D_\infty = D([0,\infty ),\mathbb{Z}^d ) $$