Measure Concentration on Fermi Balls

TL;DR

This paper investigates the conditions under which probability measures linked to an orthonormal basis in Paley-Wiener spaces converge to a uniform distribution on a compact set, with partial results suggesting convergence to a Fermi ball under certain conditions.

Contribution

It explores the convergence behavior of scaled probability measures in Paley-Wiener spaces and provides partial results indicating convergence to a Fermi ball under specific assumptions.

Findings

Indications of weak convergence to a Fermi ball for certain symbols and domains.

Results are preliminary and highlight the need for further investigation.

Generalized Weyl law conditions influence measure convergence.

Abstract

This note reports on some attempts to examine if and under which conditions the naturally scaled probability measures associated to an orthonormal basis of a classical Paley-Wiener space converge to a uniform distribution (on a compact set in momentum space). The results are still quite unsatisfactory, yet we got some indications that for inf-compact functions (symbols) and domains for which some generalized Weyl law holds, the measures converge weakly to a (generalized) Fermi ball.