Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 171, Issue 5, Pages 768-801Publisher
SPRINGER
DOI: 10.1007/s10955-018-2029-6
Keywords
Random matrix theory and extensions; Non-interacting fermions; Quantum boundary conditions; Determinantal processes; Group heat kernel; Non-intersecting paths
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Funding
- ERC Advanced Grant [669306]
- Italian National Group of Mathematical Physics (GNFM-INdAM)
- EPSRC [EP/L010305/1]
- European Research Council (ERC) [669306] Funding Source: European Research Council (ERC)
- EPSRC [EP/L010305/1] Funding Source: UKRI
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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.
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