Exact asymptotics of the freezing transition of a logarithmically correlated random energy model

TL;DR

This paper analyzes the asymptotic behavior of a logarithmically correlated random energy model, revealing insights into the freezing transition and its connections to branching processes and extreme value statistics.

Contribution

It provides a rigorous proof of the asymptotics of the partition function's generating function for the model, extending Bramson's work to a discrete-time setting.

Findings

Asymptotic properties of the generating function are characterized.

Connections to extreme value statistics of branching random walks are established.

Insights into the freezing transition of the model are provided.

Abstract

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation - thus translating Bramson's work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point.