Abstract
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.
Original language | English |
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Pages (from-to) | 397-440 |
Number of pages | 44 |
Journal | Communications in Mathematical Physics |
Volume | 169 |
Issue number | 2 |
DOIs | |
Publication status | Published - 1995 |