Fluctuations of the winding number of a directed polymer in a random medium

TL;DR

This paper derives exact formulas for the fluctuations of the winding number of a directed polymer in a random medium on an infinite cylinder, considering thermal noise and disorder variations.

Contribution

It provides the first exact expressions for the winding number fluctuations in a 1+1 dimensional directed polymer model with periodic boundary conditions.

Findings

Exact expressions for thermal fluctuation effects.

Exact expressions for disorder-induced fluctuations.

Insights into the statistical behavior of polymer winding numbers.

Abstract

For a directed polymer in a random medium lying on an infinite cylinder, that is in 1+1 dimensions with finite width and periodic boundary conditions on the transverse direction, the winding number is simply the algebraic number of turns the polymer does around the cylinder. This paper presents exact expressions of the fluctuations of this winding number due to, first, the thermal noise of the system and, second, the different realizations of the disorder in the medium.