Article

Determinantal point processes

Pierre van Moerbeke

in The Oxford Handbook of Random Matrix Theory

Published in print September 2015 | ISBN: 9780198744191
Published online August 2018 | e-ISBN: 9780191873997 | DOI: https://dx.doi.org/10.1093/oxfordhb/9780198744191.013.11

Series: Oxford Handbooks

Determinantal point processes

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This article presents a list of algebraic, combinatorial, and analytic mechanisms that give rise to determinantal point processes. Determinantal point processes have been used in random matrix theory (RMT) since the early 1960s. As a separate class, determinantal processes were first used to model fermions in thermal equilibrium and the term ‘fermion’ point processes were adopted. The article first provides an overview of the generalities associated with determinantal point processes before discussing loop-free Markov chains, that is, the trajectories of the Markov chain do not pass through the same point twice almost surely. It then considers the measures given by products of determinants, namely, biorthogonal ensembles. An especially important subclass of biorthogonal ensembles consists of orthogonal polynomial ensembles. The article also describes L-ensembles, a general construction of determinantal point processes via the Fock space formalism, dimer models, uniform spanning trees, Hermitian correlation kernels, and Pfaffian point processes.

Keywords: determinantal point process; random matrix theory (RMT); loop-free Markov chain; biorthogonal ensemble; L-ensemble; Fock space; dimer; uniform spanning tree; Hermitian correlation kernel; Pfaffian point process

Article.  10158 words. 

Subjects: Mathematics

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