Fluctuations of partition functions of directed polymers in weak disorder beyond the L2-phase

We study the directed polymer model in a bounded environment in weak disorder without L2-boundedness, specifically the speed of homogenization for the field (Wnx)x∈Zd, where Wnx denotes the associated martingale for the polymer starting from x. We show that a suitably recentered spatial average over a set of diameter n1/2 convergence to zero at rate n−ξ+o(1), where the exponent is an explicit function of the inverse temperature β.