Scaling for a random polymer

Let Q~ be the law of the n-step random walk on ~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 fl E (0,co) there exist 0*(fl)C (0,1) and #~ E {# C /I(N) : II#]lll= 1,# > 0} such that under the law Q~ as n --~ oo: (i) O*(fl) is the limit empirical speed of the random walk; (ii) #~ is the limit empirical distribution of the local times. A representation was given for 0*(fl) and #~ 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 fl ~ 0: (i) /~-i89 ~ b*; __.1 , (ii) fl 3#B(r.fl-i89 q . ( . ) . The limits b* c (0,oo) and q* E {~/E LI(IR +) " [IqliL1 = 1,q > 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(IR+). Our scaling result shows that the speed of soft polymers in d -- 1 is not right-differentiable at fl = 0, which precludes expansion techniques that have been used successfully in d >_- 5 (Hara and Slade (1992a, b)). In simulations the scaling limit is seen for fl <10 -2.