New bounds for the free energy of directed polymers in dimension 1+1 and 1+2

TL;DR

This paper advances understanding of directed polymers in random environments by providing sharp free energy estimates in 1+1 dimensions and confirming very strong disorder at all temperatures in 1+2 dimensions, solving longstanding conjectures.

Contribution

It offers new bounds for free energy in 1+1 dimensions and proves very strong disorder at all temperatures in 1+2 dimensions, resolving key open problems.

Findings

Sharp estimates on free energy at high temperature in 1+1 dimensions

Very strong disorder holds at all temperatures in 1+2 dimensions

Resolved a long-standing conjecture in the field

Abstract

We study the free energy of the directed polymer in random environment in dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and Vargas concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension 2, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.