Intermediate disorder limits for multi-layer semi-discrete directed polymers

TL;DR

This paper proves that the multi-layer semi-discrete directed polymer's partition function converges to the continuum polymer's partition function in the intermediate disorder regime, confirming a key conjecture and revealing new structural properties.

Contribution

It establishes the convergence of the semi-discrete model to the continuum model, verifying a conjecture and connecting the KPZ line ensemble to the continuum partition function.

Findings

Convergence of partition functions in the intermediate disorder regime.

Identification of the KPZ line ensemble as logarithms of ratios of layers.

Properties like continuity and positivity of the continuum partition function.

Abstract

We show that the partition function of the multi-layer semi-discrete directed polymer converges in the intermediate disorder regime to the partition function for the multi-layer continuum polymer introduced by O'Connell and Warren. This verifies, modulo a previously hidden constant, an outstanding conjecture proposed by Corwin and Hammond. A consequence is the identification of the KPZ line ensemble as logarithms of ratios of consecutive layers of the continuum partition function. Other properties of the continuum partition function, such as continuity, strict positivity and contour integral formulas to compute mixed moments, are also identified from this convergence result.