A Variational Characterization of the Speed of a One-Dimensional Self- Repellent Random Walk

Let Q.n be the probability measure for an n-step random walk (0,S1,.,Sn) on Z obtained by weighting simple random walk with a factor 1.. for every self-intersection. This is a model for a one-dimensional polymer. We prove that for every ..(0,1) there exists ..(.).(0,1) such that limn..Q.n(|Sn|n.[..(.)..,..(.)+.])=1for every.>0. We give a characterization of ..(.) in terms of the largest eigenvalue of a one-parameter family of N.N matrices. This allows us to prove that ..(.) is an analytic function of the strength . of the self-repellence. In addition to the speed we prove a limit law for the local times of the random walk. The techniques used enable us to treat more general forms of self-repellence involving multiple intersections. We formulate a partial differential inequality that is equivalent to ....(.) being (strictly) increasing. The verification of this inequality remains open.