Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality
Nagao, Taro
Akemann, Gernot
Kieburg, Mario
Parra, Iván
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 $\beta=2$
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. 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.
2019
info:eu-repo/semantics/preprint
doc-type:preprint
text
https://pub.uni-bielefeld.de/record/2936503
Nagao T, Akemann G, Kieburg M, Parra I. Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality. <em>arXiv:1905.07977</em>. 2019.
eng
info:eu-repo/semantics/altIdentifier/arxiv/1905.07977
info:eu-repo/semantics/closedAccess