# Free fermions and the classical compact groups

## TL;DR

This paper explores the connection between free fermions with various boundary conditions and eigenvalue statistics of classical compact groups, extending to finite temperatures and constructing related matrix models.

## Contribution

It introduces a comprehensive framework linking quantum boundary conditions, finite temperature effects, and eigenvalue statistics, including new matrix models for classical groups.

## Key findings

- Unified analysis of bulk and edge scaling limits
- Construction of finite temperature extensions of Haar measures
- Identification of eigenvalue statistics with grand canonical fermion models

## Abstract

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of non-interacting free fermions with classical boundary conditions.