Complex Random Energy Model: Zeros and Fluctuations

TL;DR

This paper investigates the asymptotic behavior and zeros of the partition function in the complex plane for the random energy model, providing rigorous results that extend physics predictions.

Contribution

It characterizes the asymptotic structure of zeros and proves limit theorems for the complex partition function, including correlated exponential sums.

Findings

Identification of the asymptotic zero distribution

Limit theorems for the partition function in complex domain

Extension to correlated exponential sums

Abstract

The partition function of the random energy model at inverse temperature $\beta$ is a sum of random exponentials $Z_N(\beta)=\sum_{k=1}^N \exp(\beta \sqrt{n} X_k)$, where $X_1,X_2,...$ are independent real standard normal random variables (= random energies), and $n=\log N$. We study the large $N$ limit of the partition function viewed as an analytic function of the complex variable $\beta$. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $\beta$, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.