Directed polymer in random environment and last passage percolation

TL;DR

This paper establishes a large deviations principle for the distribution of path weights in directed polymers within random environments, linking it to the free energy and analyzing path count asymptotics.

Contribution

It introduces a large deviations framework for the probability measures of path weights, connecting it to the free energy of directed polymers in random environments.

Findings

Large deviations principle for the measures $ u_n$

Rate function as Legendre transform of free energy

Asymptotic behavior of the number of high-weight paths

Abstract

The sequence of random probability measures $\nu_n$ that gives a path of length $n$, $\unsur{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.