Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment

TL;DR

This paper establishes exponential deviation inequalities for martingales and applies them to analyze the free energy of directed polymers in random environments, revealing convergence rates and limit expressions.

Contribution

It introduces new exponential inequalities for martingales and applies them to derive properties of the free energy in directed polymers, linking to multiplicative cascades.

Findings

Exponential large deviation inequalities for martingales.

Concentration inequalities for the free energy of polymers.

Expressions for the limit value of free energy in terms of cascades.

Abstract

The objective of the present paper is to establish exponential large deviation inequalities, and to use them to show exponential concentration inequalities for the free energy of a polymer in general random environment, its rate of convergence, and an expression of its limit value in terms of those of some multiplicative cascades.