Intermediate Disorder regime for half-space directed polymers

TL;DR

This paper studies the intermediate disorder regime for half-space directed polymers in 1+1 dimensions, showing convergence of partition functions to stochastic heat equation solutions with Robin boundary conditions, extending results from full space models.

Contribution

It establishes the convergence of partition functions and endpoint densities for half-space directed polymers in the intermediate disorder regime, including applications to the log-gamma polymer model.

Findings

Partition functions converge to stochastic heat equation solutions with Robin boundary conditions.

Endpoint density convergence in the intermediate disorder regime.

Application of results to the log-gamma directed polymer model.

Abstract

We consider the convergence of partition functions and endpoint density for the half-space directed polymer model in dimension $1+1$ in the intermediate disorder regime as considered for the full space model by Alberts, Khanin and Quastel in [AKQ]. By scaling the inverse temperature like $\beta n^{-1/4}$, the point-to-point partition function converges to the chaos series for the solution to stochastic heat equation with Robin boundary condition and delta initial data. We also apply our convergence results to the exact-solvable log-gamma directed polymer model in a half-space.