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∫<sup>t0δ0（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 for β 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.
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∫t0δ0(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 for β 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 Z^d. 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.
基金
supported by National Science Foundation of USA (Grant Nos. DMS1007176 and DMS-0706928)
