The glassy phase of the complex branching Brownian motion energy model

TL;DR

This paper analyzes the fluctuations of the partition function in complex branching Brownian motion energy models within the glassy phase, extending previous work to include correlated real and imaginary parts and complex temperatures.

Contribution

It provides weak limit theorems for the partition function in the glassy phase, allowing for arbitrary correlations, thus generalizing prior uncorrelated models.

Findings

Weak limit theorems established for the partition function

Extension to correlated real and imaginary energies

Includes real-valued BBM at complex temperatures

Abstract

We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase" -- the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas, where the uncorrelated case was treated. In particular, our result covers the case of the real-valued BBM energy model at complex temperatures.