On a Factorization Formula for the Partition Function of Directed Polymers

We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice \(\mathbb {Z}^{d+1}\). The polymers are subject to a random potential induced by independent identically distributed random variables and we consider the regime of weak disorder, where polymers behave diffusively. We show that when writing the quotient of the point-to-point partition function and the transition probability for the underlying random walk as the product of two point-to-line partition functions plus an error term, then, for large time intervals [0, t], the error term is small uniformly over starting points x and endpoints y in the sub-ballistic regime \(\Vert x - y \Vert \le t^{\sigma }\), where \(\sigma < 1\) can be arbitrarily close to 1. This extends a result of Sinai, who proved smallness of the error term in the diffusive regime \(\Vert x - y \Vert \le t^{1/2}\). We also derive asymptotics for spatial and temporal correlations of the field of limiting partition functions.