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A solvable mixed charge ensemble on the line: global results

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Abstract

We consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2:1 in the presence of an external field. With the total charge fixed and the system held at temperature corresponding to β = 1, it is proved that the particles form a Pfaffian point process. When the external field is quadratic (the harmonic oscillator potential), we produce the explicit family of skew-orthogonal polynomials necessary to simplify the related matrix kernels. In this setting a variety of limit theorems are proved on the distribution of the number as well as the spatial density of each species of particle as the total charge increases to infinity. Connections to Ginibre’s real ensemble of random matrix theory are highlighted throughout.

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Correspondence to Brian Rider.

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Rider, B., Sinclair, C.D. & Xu, Y. A solvable mixed charge ensemble on the line: global results. Probab. Theory Relat. Fields 155, 127–164 (2013). https://doi.org/10.1007/s00440-011-0394-z

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Keywords

  • Random matrix
  • Eigenvalue statistics
  • Pfaffian processes

Mathematics Subject Classification (2000)

  • 60B20
  • 82B05