# Solvable Particle Models Related to the Beta-Ensemble

## Datum

2013-10-03

## Autor:innen

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## Verlag

University of Oregon

## Zusammenfassung

For beta > 0, the beta-ensemble corresponds to the joint probability density on the real line proportional to
prod_{n > m}^N abs{x_n - x_m}^beta prod_{n = 1}^N w(x_n)
where w is the weight of the system. It has the application of being the Boltzmann factor for the configuration of N charge-one particles interacting logarithmically on an infinite wire inside an external field Q = -log w at inverse temperature beta. Similarly, the circular beta-ensemble has joint probability density proportional to
prod_{n > m}^N abs{e^{itheta_n} - e^{itheta_m}}^beta prod_{n = 1}^N w(x_n) quad for theta_n in [- pi, pi)
and can be interpreted as N charge-one particles on the unit circle interacting logarithmically with no external field. When beta = 1, 2, and 4, both ensembles are said to be solvable in that their correlation functions can be expressed in a form which allows for asymptotic calculations. It is not known, however, whether the general beta-ensemble is solvable.
We present four families of particle models which are solvable point processes related to the beta-ensemble. Two of the examples interpolate between the circular beta-ensembles for beta = 1, 2, and 4. These give alternate ways of connecting the classical beta-ensembles besides simply changing the values of beta. The other two examples are "mirrored" particle models, where each particle has a paired particle reflected about some point or axis of symmetry.

## Beschreibung

## SchlagwÃ¶rter

Beta Ensemble, Random Matrix Theory