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Index distribution of the Ginibre ensemble

Allez, Romain; Touboul, Jonathan; Wainrib, Gilles (2014), Index distribution of the Ginibre ensemble, Journal of Physics. A, Mathematical and Theoretical, 47, 4. http://dx.doi.org/10.1088/1751-8113/47/4/042001

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1310.5039.pdf (398.4Kb)
Type
Article accepté pour publication ou publié
Date
2014
Journal name
Journal of Physics. A, Mathematical and Theoretical
Volume
47
Number
4
Publisher
IOP Publishing
Publication identifier
http://dx.doi.org/10.1088/1751-8113/47/4/042001
Metadata
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Author(s)
Allez, Romain
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Touboul, Jonathan
Labex MemoLife
Centre interdisciplinaire de recherche en biologie [CIRB]
Wainrib, Gilles
Laboratoire Analyse, Géométrie et Applications [LAGA]
Abstract (EN)
Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. In this fast track communication, we compute analytically the probability distribution of the number of eigenvalues NR with modulus greater than R (the index) of a large N × N random matrix in the real or complex Ginibre ensemble. We show that the fraction NR/N = p has a distribution scaling as exp ( − βN2ψR(p)) with β = 2 (respectively β = 1) for the complex (resp. real) Ginibre ensemble. For any p ∈ [0, 1], the equilibrium spectral densities as well as the rate function ψR(p) are explicitly derived. This function displays a third order phase transition at the critical (minimum) value $p^*_R=1-R^2$, associated to a phase transition of the Coulomb gas. We deduce that, in the central regime, the fluctuations of the index NR around its typical value $p^*_R N$ scale as N1/3.
Subjects / Keywords
Statistical physics and nonlinear systems; Mathematical physics; Computational physics; Electronics and devices; Ginibre ensemble

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