Remarks on a constrained optimization problem for the Ginibre ensemble

TL;DR

This paper investigates the eigenvalue distribution of the Ginibre ensemble under specific constraints, using obstacle problem techniques to characterize the optimal distribution and its properties.

Contribution

It introduces a novel approach by formulating the constrained eigenvalue problem as an obstacle problem, providing new insights into the distribution's structure.

Findings

Characterizes the constrained eigenvalue distribution via obstacle problem

Establishes monotonicity properties of the optimal distribution

Provides a framework for analyzing constrained random matrix ensembles

Abstract

We study the limiting distribution of the eigenvalues of the Ginibre ensemble conditioned on the event that a certain proportion lie in a given region of the complex plane. Using an equivalent formulation as an obstacle problem, we describe the optimal distribution and some of its monotonicity properties.