Scaling for a random polymer

Abstract

LetQ ^β_n be the law of then-step random walk on ℤ^d obtained by weighting simple random walk with a factore ^−β for every self-intersection (Domb-Joyce model of “soft polymers”). It was proved by Greven and den Hollander (1993) that ind=1 and for every β∈(0, ∞) there exist θ^*(β)∈(0,1) and

such that under the lawQ ^β_n asn→∞:

$$\begin{array}{l} (i) \theta ^* (\beta ) is the \lim it empirical speed of the random walk; \\ (ii) \mu _\beta ^* is the limit empirical distribution of the local times. \\ \end{array}$$

A representation was given forθ ^*(β) andµ ^β_β 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 β⇂0:

$$\begin{array}{l} (i) \beta ^{ - {\textstyle{1 \over 3}}} \theta ^* (\beta ) \to b^* ; \\ (ii) \beta ^{ - {\textstyle{1 \over 3}}} \mu _\beta ^* \left( {\left\lceil { \cdot \beta ^{ - {\textstyle{1 \over 3}}} } \right\rceil } \right) \to ^{L^1 } \eta ^* ( \cdot ) . \\ \end{array}$$

The limitsb ^*∈(0, ∞) and

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 onL ²(ℝ⁺). Our scaling result shows that the speed of soft polymers ind=1 is not right-differentiable at β=0, which precludes expansion techniques that have been used successfully ind≧5 (Hara and Slade (1992a, b)). In simulations the scaling limit is seen for β≦10⁻².