Dynamic polymers: invariant measures and ordering by noise

TL;DR

This paper introduces a dynamical framework for infinite volume directed polymers in random environments, proving existence, uniqueness, and ordering properties, and establishing a connection between polymer dynamics and invariant measures.

Contribution

It develops a stochastic gradient flow approach for polymers, proving key properties and linking dynamics to static measures, including the One Force -- One Solution principle.

Findings

Existence and uniqueness of polymer dynamics solutions.

Order-preserving property with respect to initial conditions.

Unique invariant measure for fixed slopes and environments.

Abstract

We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow on polymers pinned at the origin, for energy involving quadratic nearest neighbor interaction and local interaction with random environment. We prove existence and uniqueness of the solution, continuity of the flow, the order-preserving property with respect to the coordinatewise partial order, and the invariance of the asymptotic slope. We establish ordering by noise which means that if two initial conditions have distinct slopes, then the associated solutions eventually get ordered coordinatewise. This, along with the shear-invariance property and existing results on static infinite volume polymer measures, allows to prove that for a fixed asymptotic slope and almost every realization of the environment, the polymer dynamics has a unique invariant distribution given by a unique infinite volume polymer measure, and, moreover, One Force -- One Solution principle holds. We also prove that every polymer measure is concentrated on paths with well-defined asymptotic slopes and give an estimate on deviations from straight lines.