Finite-Rank Complex Deformations of Random Band Matrices: Sigma-Model Approximation
DOI:
https://doi.org/10.15407/mag19.01.211Анотація
Ми вивчаємо розподiл комплексних власних значень $z_1, \ldots , z_N$ випадкової ермiтової блокової стрiчкової матрицi розмiру N×N з комплексною деформацiєю скiнченного рангу. У режимi, коли розмiр блоков $W$ зростає швидше за $\sqrt{N}$, ми доводимо, що гранична щiльнiсть $\Im z_1, \ldots , \Im z_N$ у сiгма-модельнiй апроксимацiї збiгається з вiдповiдною щiльнiстю для Ґаусiвського унiтарного ансамблю. Для цього ми використовуємо метод, розроблений в [16].
Mathematical Subject Classification 2020: 60B20
Ключові слова:
випадковi стрiчковi матрицi, делокалiзований режим, комплексна деформацiя, сiгма-модель, суперсиметрiяПосилання
F.A. Berezin, Introduction to the algebra and analysis of anticommuting variables, Moscow State University Publ., Moscow, 1983 (Russian).
L.V. Bogachev, S.A. Molchanov, and L.A. Pastur, On the level density of random band matrices, Mat. Zametki, 50:6 (1991), 31--42. https://doi.org/10.1007/BF01158263
P. Bourgade, Random band matrices, Proc. Int. Cong. of Math., Rio de Janeiro 3 (2018), 2745--2770.
K.B. Efetov and A.I. Larkin, Kinetics of a quantum particle in a long metallic wire, Sov. Phys. JETP 58 (2) (1983), 444--451.
Y.V. Fyodorov, Negative moments of characteristic polynomials of random matrices: Ingham--Siegel integral as an alternative to Hubbard--Stratonovich transformation, Nuclear Phys. B 621 (2002), 643--674. https://doi.org/10.1016/S0550-3213(01)00508-9
Y.V. Fyodorov and B. A. Khoruzhenko, Systematic analytical approach to correlation functions of resonances in quantum chaotic scattering, Phys. Rev. Lett. 83 (1999), 65--68. https://doi.org/10.1103/PhysRevLett.83.65
Y.V. Fyodorov and A.D. Mirlin, Scaling properties of localization in random band matrices: a σ-model approach, Phys. Rev. Lett. 67 (1991), 2405--2409. https://doi.org/10.1103/PhysRevLett.67.2405
Y.V. Fyodorov and M. Osman, Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities, preprint, https://arxiv.org/abs/2105.03665
Y.V. Fyodorov and D. Savin, Resonance scattering of waves in chaotic systems, The Oxford handbook of random matrix theory (Eds. G.~Akemann et al.), Oxford University Press, 2011, 703--722.
Y.V. Fyodorov and H.-J. Sommers, Statistics of S-matrix poles in few-channel chaotic scattering: crossover from isolated to overlapping resonances, Jetp Lett. 63 (1996), 1026--1030. https://doi.org/10.1134/1.567120
Y.V. Fyodorov and H.-J. Sommers, Random matrices close to Hermitian or unitary: overview of methods and results, J.Phys.A: Math.Gen. 36 (2003), 3303--3347. https://doi.org/10.1088/0305-4470/36/12/326
Y.V. Fyodorov, M.A. Skvortsov, and K.S. Tikhonov, Resonances in a single-lead reflection from a disordered medium: σ-model approach, https://arxiv.org/abs/2211.03376
R. Kozhan, Rank one non-Hermitian perturbations of Hermitian beta-ensembles, J. Stat. Phys. 168: 1 (2017), 92--108. https://doi.org/10.1007/s10955-017-1792-0
G.E. Mitchell, A. Richter, and H.A. Weidenmüller, Random matrices and chaos in nuclear physics: nuclear reactions, Rev. Modern Phys. 82:4 (2010), 2845--2901. https://doi.org/10.1103/RevModPhys.82.2845
S. A. Molchanov, L.A. Pastur, and A.M. Khorunzhii, Distribution of the eigenvalues of random band matrices in the limit of their infinite order, Theor. Math. Phys. 90 (1992), 108--118. https://doi.org/10.1007/BF01028434
M. Shcherbina and T. Shcherbina, Universality for 1d random band matrices: sigma-model approximation J. Stat. Phys, 172:2 (2018), 627--664. https://doi.org/10.1007/s10955-018-1969-1
M. Shcherbina and T. Shcherbina, Universality for 1 d random band matrices, Commun. Math. Phys. 385 (2021), 667 -- 716 https://doi.org/10.1007/s00220-021-04135-6
V.V. Sokolov and V.G. Zelevinsky, Dynamics and statistics of unstable quantum states, Nucl. Phys. A 504 (1989), 562--588. https://doi.org/10.1016/0375-9474(89)90558-7
H.-J. Sommers, Y.V. Fyodorov, and M. Titov, S-matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices: from isolated to overlapping resonances, J. Phys. A 32 (1999), No. 5, L77--L87. https://doi.org/10.1088/0305-4470/32/5/003
J.J.M. Verbaarschot, H.A. Weidenmüller, and M.R. Zirnbauer, Grassmann integrations and stochastic quantum physics, Phys. Rep. 129 (1985), 367--438. https://doi.org/10.1016/0370-1573(85)90070-5
F.J. Wegner, Disordered system with $n$ orbitals per site: $n$ ➝ ꝏ limit, Phys. Rev. B 19 (1979),783--792. https://doi.org/10.1103/PhysRevB.19.783
