A Generalized Plasma and Interpolation Between Classical Random Matrix Ensembles

Abstract

The eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log-gas at the special couplings β=1,2 and 4. It has been known for some time that there is an exactly solvable two-component log-potential plasma which interpolates between the β=1 and 4 circular ensemble, and an exactly solvable two-component generalized plasma which interpolates between β=2 and 4 circular ensemble. We extend known exact results relating to the latter—for the free energy and one and two-point correlations—by giving the general (k 1+k 2)-point correlation function in a Pfaffian form. Crucial to our working is an identity which expresses the Vandermonde determinant in terms of a Pfaffian. The exact evaluation of the general correlation is used to exhibit a perfect screening sum rule.

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Correspondence to Christopher D. Sinclair.

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Forrester, P.J., Sinclair, C.D. A Generalized Plasma and Interpolation Between Classical Random Matrix Ensembles. J Stat Phys 143, 326–345 (2011). https://doi.org/10.1007/s10955-011-0173-3

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Keywords

  • Random matrix
  • Generalized plasma
  • Log-gas
  • Pfaffian
  • Two-point correlation
  • Anomalous quantum Hall effect