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Castillo IP, Guzmán-González E, Sánchez ATR, Metz FL. Analytic approach for the number statistics of non-Hermitian random matrices. Phys Rev E 2021; 103:062108. [PMID: 34271724 DOI: 10.1103/physreve.103.062108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2020] [Accepted: 05/18/2021] [Indexed: 11/07/2022]
Abstract
We introduce a powerful analytic method to study the statistics of the number N_{A}(γ) of eigenvalues inside any smooth Jordan curve γ∈C for infinitely large non-Hermitian random matrices A. Our generic approach can be applied to different random matrix ensembles of a mean-field type, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable, and obtain explicit results for the diluted real Ginibre ensemble. The main outcome is an effective theory that determines the cumulant generating function of N_{A} via a path integral along γ, with the path probability distribution following from the numerical solution of a nonlinear self-consistent equation. We derive expressions for the mean and the variance of N_{A} as well as for the rate function governing rare fluctuations of N_{A}(γ). All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.
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Affiliation(s)
- Isaac Pérez Castillo
- Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, Mexico
| | - Edgar Guzmán-González
- Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, Mexico
| | | | - Fernando L Metz
- Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil.,London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom
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Smith NR, Le Doussal P, Majumdar SN, Schehr G. Counting statistics for noninteracting fermions in a d-dimensional potential. Phys Rev E 2021; 103:L030105. [PMID: 33862753 DOI: 10.1103/physreve.103.l030105] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 03/03/2021] [Indexed: 11/07/2022]
Abstract
We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.
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Affiliation(s)
- Naftali R Smith
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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Lacroix-A-Chez-Toine B, Garzón JAM, Calva CSH, Castillo IP, Kundu A, Majumdar SN, Schehr G. Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble. Phys Rev E 2019; 100:012137. [PMID: 31499884 DOI: 10.1103/physreve.100.012137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2019] [Indexed: 06/10/2023]
Abstract
We study the Ginibre ensemble of N×N complex random matrices and compute exactly, for any finite N, the full distribution as well as all the cumulants of the number N_{r} of eigenvalues within a disk of radius r centered at the origin. In the limit of large N, when the average density of eigenvalues becomes uniform over the unit disk, we show that for 0<r<1 the fluctuations of N_{r} around its mean value 〈N_{r}〉≈Nr^{2} display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order O(N^{1/4}), (ii) an intermediate regime where N_{r}-〈N_{r}〉=O(sqrt[N]), and (iii) a large deviation regime where N_{r}-〈N_{r}〉=O(N). This intermediate behavior (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centered) cumulants of N_{r}, which are all of order O(sqrt[N]). We show that the intermediate deviation function that describes these intermediate fluctuations can be computed explicitly and we demonstrate that it is universal, i.e., it holds for a large class of complex random matrices. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.
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Affiliation(s)
| | - Jeyson Andrés Monroy Garzón
- Departamento de Física Cuántica y Fotónica, Instituto de Física, UNAM, P.O. Box 20-364, 01000 Mexico Distrito Federal, Mexico
| | | | - Isaac Pérez Castillo
- Departamento de Física Cuántica y Fotónica, Instituto de Física, UNAM, P.O. Box 20-364, 01000 Mexico Distrito Federal, Mexico
- London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom
| | - Anupam Kundu
- International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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Pérez Castillo I, Metz FL. Theory for the conditioned spectral density of noninvariant random matrices. Phys Rev E 2018; 98:020102. [PMID: 30253505 DOI: 10.1103/physreve.98.020102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2018] [Indexed: 06/08/2023]
Abstract
We develop a theoretical approach to compute the conditioned spectral density of N×N noninvariant random matrices in the limit N→∞. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction k of eigenvalues smaller than x∈R, provides the spectrum of random matrix samples that deviate atypically from the average behavior. We apply our theory to sparse random matrices and unveil strikingly different and generic properties, namely, (i) their conditioned spectral density has compact support, (ii) it does not experience any abrupt transition for k around its typical value, and (iii) its eigenvalues do not accumulate at x. Moreover, our work points towards other types of transitions in the conditioned spectral density for values of k away from its typical value. These properties follow from the weak or absent eigenvalue repulsion in sparse ensembles and they are in sharp contrast to those displayed by classic or rotationally invariant random matrices. The exactness of our theoretical findings are confirmed through numerical diagonalization of finite random matrices.
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Affiliation(s)
- Isaac Pérez Castillo
- Department of Quantum Physics and Photonics, Institute of Physics, UNAM, P.O. Box 20-364, 01000 Mexico City, Mexico and London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom
| | - Fernando L Metz
- Institute of Physics, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil; Physics Department, Federal University of Santa Maria, 97105-900 Santa Maria, Brazil; and London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom
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