Scaling for a random polymer

LetQ _n ^β be the law of then-step random walk on ?^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 _n ^β 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 ? x ? matrices. In the present paper we use this representation to prove the following scaling result as β?0: $$begin{array}{l} (i) beta ^{ - {textstyle{1 over 3}}} theta ^* (beta ) to b^* ; (ii) beta ^{ - {textstyle{1 over 3}}} mu _beta ^* left( {leftlceil { cdot beta ^{ - {textstyle{1 over 3}}} } rightrceil } right) to ^{L^1 } eta ^* ( cdot ) . end{array}$$ 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(?⁺). 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.