# Statistics of fermions in a $d$-dimensional box near a hard wall

## TL;DR

This paper investigates the universal statistical behavior of noninteracting fermions near a hard wall boundary in any dimension, revealing a new kernel and extending understanding of edge phenomena in quantum gases.

## Contribution

It introduces a new universal edge kernel for fermions near hard walls in arbitrary dimensions and computes it explicitly for spherical domains.

## Key findings

- Derived the hard edge kernel for fermions in a spherical domain.
- Showed the kernel's universality near smooth boundaries using a generalized method of images.
- Extended results to non-smooth boundaries and finite temperature scenarios.

## Abstract

We study $N$ noninteracting fermions in a domain bounded by a hard wall potential in $d \geq 1$ dimensions. We show that for large $N$, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth potentials. We compute this $d$ dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary. As an application we compute the quantum statistics of the position of the fermion closest to the wall. Our results are then extended in several directions, including non-smooth boundaries such as a wedge, and also to finite temperature.