Concentration results for directed polymer with unbounded jumps

TL;DR

This paper investigates the concentration properties of free energy and ground state energy for directed polymers with unbounded jumps in a Bernoulli environment, revealing high-probability bounds and properties relevant to first passage percolation.

Contribution

It establishes concentration results for the ground state energy and free energy of directed polymers with unbounded jumps in a Bernoulli environment, a novel extension in the field.

Findings

Concentration of ground state energy at zero temperature

Concentration of free energy at finite temperature

Maximum jump size of near-minimizing polymers is bounded with high probability

Abstract

We study the free energy and its relevant quantity for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by the Bernoulli variables. We first establish the concentration of the ground state energy of polymer at zero temperature. Secondly, we also prove the same property of the free energy at finite temperature. In the proof, we use the fact that the maximum jump of any polymer nearly minimizing energy is not too large with high probability. This is an interesting property itself from the first passage percolation viewpoint.