Free Fermions and the Classical Compact Groups

Fabio Deelan Cunden*, Francesco Mezzadri, Neil O’Connell

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

17 Citations (Scopus)
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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.

Original languageEnglish
Pages (from-to)768-801
Number of pages34
JournalJournal of Statistical Physics
Issue number5
Early online date20 Apr 2018
Publication statusPublished - Jun 2018


  • Determinantal processes
  • Group heat kernel
  • Non-interacting fermions
  • Non-intersecting paths
  • Quantum boundary conditions
  • Random matrix theory and extensions

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