Abstract
There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: (i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; (ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions.
Highlights
In this paper we introduce and discuss several extensions of the eigenvalue statistics induced by the Haar measure on the classical compact groups U(2N + 1), Sp(2N ), SO(2N + 1), and SO(2N )
By considering all the admissible boundary conditions, we show that the processes defined by the Haar measure on the classical compact groups are immersed in a four-parameter family of determinantal processes associated to free fermions in a box
We show that these determinantal processes interpolate between random matrix and Poisson statistics and we investigate the simultaneous limit of high temperature and large number of particles
Summary
In this paper we introduce and discuss several extensions of the eigenvalue statistics induced by the Haar measure on the classical compact groups U(2N + 1), Sp(2N ), SO(2N + 1), and SO(2N ). The starting point of this work is the following connection between the classical compact groups and free fermions in the ground state: The eigenvalues of random matrices sampled according to the Haar measure on the classical compact groups, and the particle density of free (non-interacting) fermions in a box with classical boundary conditions at zero temperature, form the same determinantal point processes. This follows from well known formulae for the joint law of eigenvalues of random matrices, and elementary diagonalisation of Schrödinger operators. These new ensembles of random matrices are constructed by i) ‘evolving’ the Haar measure along the heat flow on the classical compact groups, and ii) by considering a suitable randomization on the size of the group (grand canonical construction)
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