The Volume of the Trace-Nonnegative Polytope via the Irwin-Hall Distribution

TL;DR

This paper derives an explicit formula for the volume of the trace nonnegative polytope using a generalized Irwin-Hall distribution, providing bounds for spectra and suggesting future research directions.

Contribution

It introduces a novel explicit volume formula for the trace nonnegative polytope based on a generalized Irwin-Hall distribution, advancing spectral geometry understanding.

Findings

Derived an explicit volume formula for the trace nonnegative polytope.

Established that this volume bounds the space of realizable spectra.

Presented auxiliary results on trace-zero spectra and posed open problems.

Abstract

In this work, we find an explicit expression for the volume of the trace nonnegative polytope via a generalization of the Irwin-Hall distribution. This volume is an upper bound for the volume of all projected, normalized realizable spectra. We provide ancillary results on realizable trace-zero spectra and pose several problems suitable for further inquiry.