Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

Abstract

We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature β = 2 we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all k-point correlation functions for a fixed number of particles N . These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-N limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine-and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

“…For comparison, "pointwise" root-type singularities typically produce other kinds of ingredients in the asymptotics, such as Barnes' G-function (as was discovered by Basor [6] in dimension one and by Webb and Wong [61] in dimension two), and "circular" jump-type singularities involve the error function. We also mention that ensembles with "circular" root-type singularities have been studied in [64,55], and ensembles with "elliptic" root-type singularities in [52]. In [64,52,55], the singularities are located at the hard edge and the focus was on the leading order behavior of the kernel; in particular, the (associated) Hermite polynomials do not show up in these works.…”

Random normal matrices in the almost-circular regime

“…[4,9]. We remark that in [34], the authors considered another notion of hard edge cuts, which yield different scaling limits.…”

Wronskian structures of planar symplectic ensembles

“…From a statistical physics point of view, this confinement has the effect of condensing a non-trivial portion of the particles onto the hard edge. We refer to [34,41,46] and references therein for the studies of complex ensembles (1.10) with such type of boundary confinement. (See also [52] for a similar situation in the context of truncated unitary ensembles.…”