On the partition function of a directed polymer in a Gaussian random environment

Abstract

The purpose of this work is the study of the partition function Zn(β) of a (d+1)-dimensional lattice directed polymer in a Gaussian random environment (β > 0 being the inverse of temperature). In the low-dimensional cases (d = 1 or d = 2), we prove that for all β > 0, the renormalized partition function Zn (β)/EZn(β) converges to 0 and the correlation 〈1(Sn1=Sn2)〉(n) of two independent configurations does not converge to 0. In the high dimensional case (d ≥ 3), a lower tail of Zn(β) has been obtained for small β > 0. Furthermore, we express some thermodynamic quantities in terms of the path measure alone.