SCALING FOR A RANDOM POLYMER

Let Q(n)beta be the law of the n-step random walk on Z(d) obtained by weighting simple random walk with a factor e(-beta) 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 beta is-an-element-of (0, infinity) there exist theta* (beta) is-an-element-of (0,1) and mu(beta)* is-an-element-of {mu is-an-element-of l1 (N): parallel-to mu parallel-to (l1) = 1, mu > 0} such that under the law Q(n)beta as n --> infinity: (i) theta* (beta) is the limit empirical speed of the random walk; (ii) mu(beta)* is the limit empirical distribution of the local times. A representation was given for theta*(beta) and mu(beta)* in terms of a largest eigenvalue problem for a certain family of N x N matrices. In the present paper we use this representation to prove the following scaling result as beta down 0: (i) beta-1/3 theta* (beta) --> b*; (ii) beta-1/3 mu(beta)*(inverted left perpendicular . beta-1/3 inverted right-perpendicular) --> L1 eta* (.). The limits b* is-an-element-of (0, infinity) and eta* is-an-element-of {eta is-an-element-of L1 (R+): parallel-toetaparallel-to(L)1 = 1, eta > 0} are identified in terms of a Sturm-Liouville problem, which turns out to have several interesting properties. The techniques that are used in the proof are functional analytic and revolve around the notion of epi-convergence of functionals on L2(R+). Our scaling result shows that the speed of soft polymers in d = 1 is not right-differentiable at beta = 0, which precludes expansion techniques that have been used successfully in d greater-than-or-equal-to 5 (Hara and Slade (1992a, b)). In simulations the scaling limit is seen for beta less-than-or-equal-to 10(-2).