Particle Systems with Repulsion Exponent $β$ and Random Matrices

We consider a class of particle systems generalizing the $β$-Ensembles from random matrix theory. In these new ensembles, particles experience repulsion of power $β>0$ when getting close, which is the same as in the $β$-Ensembles. For distances larger than zero, the interaction is allowed to differ from those present for random eigenvalues. We show that the local bulk correlations of the $β$-Ensembles, universal in random matrix theory, also appear in these new ensembles.

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