# Spectral statistics of the uni-modular ensemble

## TL;DR

This paper studies the spectral properties of large Hermitian matrices with elements uniformly distributed on the unit circle, employing three methods to analyze spectral moments and fluctuations, and compares their effectiveness.

## Contribution

It introduces three complementary analytical methods to study the spectral statistics of the uni-modular ensemble and compares their advantages, providing new insights into spectral fluctuations.

## Key findings

- Derived expressions for 1/N corrections to spectral moments
- Analyzed fluctuations around the mean spectral moments
- Compared the effectiveness of supersymmetric, combinatorial, and Brownian motion methods

## Abstract

We investigate the spectral statistics of Hermitian matrices in which the elements are chosen uniformly from U (1), called the uni-modular ensemble (UME), in the limit of large matrix size. Using three complimentary methods; a supersymmetric integration method, a combinatorial graph-theoretical analysis and a Brownian motion approach, we are able to derive expressions for 1/N corrections to the mean spectral moments and also analyse the fluctuations about this mean. By addressing the same ensemble from three different point of view, we can critically compare their relative advantages and derive some new results.