Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field

TL;DR

This paper refines conditions for the quasi-Gibbs property of infinite-dimensional interacting Brownian motions, enabling the analysis of Airy random point fields crucial in random matrix theory.

Contribution

It provides a new sufficient condition for the quasi-Gibbs property, facilitating the study of Airy RPFs in infinite-dimensional stochastic differential equations.

Findings

Refined the quasi-Gibbs property condition for RPFs.

Applied the result to prove the property for Airy RPFs.

Enabled solving ISDEs with Airy RPFs as equilibrium states.

Abstract

We give a new sufficient condition of the quasi-Gibbs property. This result is a refinement of one given in a previous paper (\cite{o.rm}), and will be used in a forth coming paper to prove the quasi-Gibbs property of Airy random point fields (RPFs) and other RPFs appearing under soft-edge scaling. The quasi-Gibbs property of RPFs is one of the key ingredients to solve the associated infinite-dimensional stochastic differential equation (ISDE). Because of the divergence of the free potentials and the interactions of the finite particle approximation under soft-edge scaling, the result of the previous paper excludes the Airy RPFs, although Airy RPFs are the most significant RPFs appearing in random matrix theory. We will use the result of the present paper to solve the ISDE for which the unlabeled equilibrium state is the $\mathrm{Airy}_{\beta}$ RPF with $ \beta = 1,2,4 $.