Differentiability of the Shape Function for Directed Polymers in Continuous Space

Abstract

For directed polymers, the shape function computes the limiting average energy accrued by paths with a given average slope. We prove that, for a large family of directed polymer models in discrete time and continuous space in dimension $1+1$, for positive and zero temperature, the shape function is differentiable with respect to the slope on the entire real line.

Minor changes in this version, corrected a few misprints. 28 pages