Scaling for a random polymer

Abstract

Let Q n be the law of then-step random walk on Z d obtained by weighting simple random walk with a factor e for every self-intersection (Domb-Joyce model of `soft polymers'). It was proved by Greven and den Hollander (1993) that in d = 1 and for every 2 (0; 1) there exist () 2 (0; 1) and 2 f 2 l 1 ( N ) :kkl 1 = 1; >0g such that under the law Q as n !1: (i) () is the limit empirical speed of the random walk; (ii) is the limit empirical distribution of the local times. A representation was given for () and in terms of a largest eigenvalue problem for a certain family of N N matrices. In the present paper we use this representation to prove the following scaling result as # 0: (i) (i) 3 () ! B ; (ii)