On the critical behavior of a homopolymer model

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（8）
• 页码范围: 1463-1476
• 页数: 14页
• 语种: 英文

摘要

We begin with the reference measure p0 induced by simple,symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define,for β ≥ 0,t ＞ 0,the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as dPβ,t/dP0=Zβ t(0)-1eβ∫0tδ0(xs)ds,(0.1) where Zβ,t(0) ≡ E0[eβ ∫0T δ0(xs)ds].This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin.In a previous paper (Cranston and Molchanov,2007),we showed that for dimensions d ≥ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior forβ above this critical value.The critical value was determined by means of the spectral properties of the operator △ + βδ0,where △ is the discrete Laplacian on Zd.This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer.In this paper we give a description of the polymer at the critical value where the phase transition takes place.The behavior at the critical parameter is dimension-dependent.