Generic regularity of equilibrium measures for the logarithmic potential with external fields

TL;DR

This paper proves that, generically, external potentials in logarithmic potential models are regular, confirming a long-standing conjecture by linking minimizing measures to thin obstacle problems.

Contribution

It establishes the generic regularity of equilibrium measures in logarithmic potential models, providing a rigorous proof of the conjecture.

Findings

External potentials are generically off-critical.

Equilibrium measures are regular in the generic case.

Connection established between minimizing measures and thin obstacle problems.

Abstract

It is a well-known conjecture in $\beta$-models and in their discrete counterpart that, generically, external potentials should be ``off-critical'' (or, equivalently, ``regular''). Exploiting the connection between minimizing measures and thin obstacle problems, we give a positive answer to this conjecture.