Change of variables as a method to study general $\beta$-models: bulk universality

Abstract

We consider $\beta$ matrix models with real analytic potentials. Assuming that the corresponding equilibrium density $\rho$ has a one-interval support (without loss of generality $\sigma =[-2,2]$), we study the transformation of the correlation functions after the change of variables ${\lambda }_i\to \zeta ({\lambda }_i)$ with $\zeta (\lambda )$ chosen from the equation ${\zeta }^{\prime }(\lambda )\rho (\zeta (\lambda ))={\rho }_{sc}(\lambda )$, where ${\rho }_{sc}(\lambda )$ is the standard semicircle density. This gives us the "deformed" $\beta$-model which has an additional "interaction" term. Standard transformation with the Gaussian integral allows us to show that the "deformed" $\beta$-model may be reduced to the standard Gaussian $\beta$-model with a small perturbation $n^{-1}h(\lambda )$. This reduces most of the problems of local and global regimes for $\beta$-models to the corresponding problems for the Gaussian $\beta$-model with a small perturbation. In the present paper we prove the bulk universality of local eigenvalue statistics for both one-cut and multi-cut cases.

20 pages

References in corpus (4)

• Bulk Universality and Related Properties of Hermitian Matrix Models
• Fluctuations of eigenvalues of matrix models and their applications
• Universality for orthogonal and symplectic Laguerre-type ensembles
• On Universality for Orthogonal Ensembles of Random Matrices

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• Bulk eigenvalue statistics for random regular graphs
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• Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
• Quenched universality for deformed Wigner matrices
• CLT for real beta-ensembles at high temperature