Remarks on the one-point density of Hele-Shaw β-ensembles - Ameur, Yacin; Troedsson, Erik (Revista Matematica Iberoamericana, 2025)
In this note, we prove equicontinuity for the family of one-point densities with respect to a two-dimensional Coulomb gas at an inverse temperature β ≥ 1=2 confined by an external potential of Hele-Shaw (or quasi-harmonic) type. As a consequence, subsequential limiting Lipschitz continuous densities are defined on the microscopic scale. There are several additional results, for example comparing the one-point density with the thermal equilibrium density.
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Keywords: equicontinuity; Hele-Shaw potential; one-point density; thermal equilibrium measure; Ward's identity; β-ensemble