Про кореляцiйнi функцiї характеристичних полiномiв випадкових матриць з незалежними елементами: iнтерполяцiя мiж комплексним i дiйсним випадками

Автор(и)

  • Ievgenii Afanasiev B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine image/svg+xml

DOI:

https://doi.org/10.15407/mag18.02.159

Ключові слова:

теорiя випадкових матриць, ансамбль Жинiбра, кореляцiйнi функцiї характеристичних полiномiв, моменти характеристичних полiномiв, суперсиметрiя

Анотація

У роботi розглянуто кореляцiйнi функцiї характеристичних полiномiв випадкових матриць з незалежними комплексними елементами. Ми дослiдили те, як асимптотична поведiнка кореляцiйних функцiй залежить вiд другого моменту спiльного закону розподiлу ймовiрностей для матричних елементiв, при цьому другий момент можна трактувати як свого роду “мiру дiйсностi” елементiв. Показано, що кореляцiйнi функцiї ведуть себе таким же чином, як i у випадку комплексного ансамблю Жинiбра, з точнiстю до множника, що залежить лише вiд другого моменту та абсолютного четвертого моменту спiльного розподiлу ймовiрностей матричних елементiв.

Mathematical Subject Classification 2010: 60B20, 15B52

Посилання

I. Afanasiev, On the Correlation Functions of the Characteristic Polynomials of the Sparse Hermitian Random Matrices, J. Stat. Phys. 163 (2016), 324-356. https://doi.org/10.1007/s10955-016-1486-z

I. Afanasiev, On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries, J. Stat. Phys. 176 (2019), 1561-1582. https://doi.org/10.1007/s10955-019-02353-w

I. Afanasiev, On the Correlation Functions of the Characteristic Polynomials of Real Random Matrices with Independent Entries, J. Math. Phys. Anal. Geom. 16 (2020), 91-118. https://doi.org/10.15407/mag16.02.091

G. Akemann and E. Kanzieper, Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem, J. Stat. Phys. 129 (2007), 1159-1231. https://doi.org/10.1007/s10955-007-9381-2

G. Akemann and G. Vernizzi, Characteristic polynomials of complex random matrix models, Nucl. Phys. B 660 (2003), 532-556. https://doi.org/10.1016/S0550-3213(03)00221-9

Z. Bao and L. Erdős, Delocalization for a class of random block band matrices, Probab. Theory Relat. Fields 167 (2017), 673-776. https://doi.org/10.1007/s00440-015-0692-y

F.A. Berezin, Introduction to superanalysis, Number 9 in Math. Phys. Appl. Math. D. Reidel Publishing Co., Dordrecht, 1987. https://doi.org/10.1007/978-94-017-1963-6_3

C. Bordenave and D. Chafai, Around the circular law, Probab. Surv. 9 (2012), 1-89. https://doi.org/10.1214/11-PS183

A. Borodin and C.D. Sinclair, The Ginibre Ensemble of Real Random Matrices and its Scaling Limits, Comm. Math. Phys. 291 (2009), 177-224. https://doi.org/10.1007/s00220-009-0874-5

A. Borodin and E. Strahov, Averages of characteristic polynomials in random matrix theory, Comm. Pure Appl. Math. 59 (2006), 161-253. https://doi.org/10.1002/cpa.20092

E. Bratus and L. Pastur, The dynamics of quantum correlations of two qubits in a common environment, J. Math. Phys. Anal. Geom. 16 (2020), No. 3, 228-262. https://doi.org/10.15407/mag16.03.228

E. Brézin and S. Hikami, Characteristic polynomials of random matrices, Comm. Math. Phys. 214 (2000), 111-135. https://doi.org/10.1007/s002200000256

E. Brézin and S. Hikami. Characteristic polynomials of real symmetric random matrices, Comm. Math. Phys. 223 (2001), 363-382. https://doi.org/10.1007/s002200100547

G. Cipolloni, L. Erdős and D. Schröder, Optimal lower bound on the least singular value of the shifted Ginibre ensemble, Prob. Math. Physics 1 (2020), 101-146. https://doi.org/10.2140/pmp.2020.1.101

G. Cipolloni, L. Erdős and D. Schröder, Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices, Probab. Theory Related Fields 179 (2021), 1-28. https://doi.org/10.1007/s00440-020-01003-7

G. Cipolloni, L. Erdős, and D. Schröder, Fluctuation around the circular law for random matrices with real entries, Electron. J. Prob., 24 (2021), Paper No. 24. https://doi.org/10.1214/21-EJP591

G. Cipolloni, L. Erdős, and D. Schröder, Edge universality for non-Hermitian random matrices, Comm. Pure Appl. Math. (2022), DOI 10.1002/cpa.22028.

M. Disertori and M. Lager, Density of States for Random Band Matrices in Two Dimensions, Ann. Henri Poincaré 18 (2017), 2367-2413. https://doi.org/10.1007/s00023-017-0572-3

M. Disertori and M. Lager, Supersymmetric Polar Coordinates with applications to the Lloyd model, Math. Phys. Anal. Geom. 23 (1) (2020), Paper No. 2. https://doi.org/10.1007/s11040-019-9326-4

M. Disertori, M. Lohmann, and S. Sodin, The density of states of 1D random band matrices via a supersymmetric transfer operator, J. Spectr. Theory 11 (1) (2021), 125-191. https://doi.org/10.4171/JST/338

M. Disertori, F. Merkl, and S. Rolles, Localization for a Nonlinear Sigma Model in a Strip Related to Vertex Reinforced Jump Processes, Commun. Math. Phys. 332 (2014), 783-825. https://doi.org/10.1007/s00220-014-2102-1

M. Disertori, T. Spencer, and M.R. Zirnbauer, Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model, Comm. Math. Phys. 300 (2010), 435-486. https://doi.org/10.1007/s00220-010-1117-5

A. Edelman, The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60 (1997), 203-232. https://doi.org/10.1006/jmva.1996.1653

K. Efetov, Supersymmetry in disorder and chaos, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511573057

K.B. Efetov, Supersymmetry and theory of disordered metals, Adv. in Physics 32 (1983), 53-127. https://doi.org/10.1080/00018738300101531

P. Forrester and T. Nagao, Eigenvalue statistics of the real Ginibre ensemble, Phys. Rev. Lett. 99 (2007), 050603. https://doi.org/10.1103/PhysRevLett.99.050603

P.J. Forrester, Fluctuation formula for complex random matrices, J. Phys. A 32 (1999), L159-L163. https://doi.org/10.1088/0305-4470/32/13/003

Y.V. Fyodorov, Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation, Nucl. 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, Localization in ensemble of sparse random matrices, Phys. Rev. Lett. 67 (1991), 2049-2052. https://doi.org/10.1103/PhysRevLett.67.2049

Y.V. Fyodorov and H.-J. Sommers, Random matrices close to Hermitian or unitary: overview of methods and results, J. Phys. A 36 (2003), 3303-3347. https://doi.org/10.1088/0305-4470/36/12/326

Y.V. Fyodorov and E. Strahov, An exact formula for general spectral correlation function of random Hermitian matrices. Random matrix theory, J. Phys. A 36 (2003), 3203-3214. https://doi.org/10.1088/0305-4470/36/12/320

J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6 (1965), 440-449. https://doi.org/10.1063/1.1704292

V.L. Girko, The circular law, Teor. Veroyatn. Primen. 29 (1984), 669-679.

V.L. Girko, The circular law: ten years later, Random Oper. Stoch. Equ. 2 (1994), 235-276. https://doi.org/10.1515/rose.1994.2.3.235

V.L. Girko, The strong circular law. Twenty years later. I, Random Oper. Stoch. Equ. 12 (2004), 49-104. https://doi.org/10.1515/156939704323067834

V.L. Girko, The strong circular law. Twenty years later. II, Random Oper. Stoch. Equ. 12 (2004), 255-312. https://doi.org/10.1515/1569397042222477

V.L. Girko, The circular law. Twenty years later. III, Random Oper. Stoch. Equ. 13 (2005), 53-109. https://doi.org/10.1515/1569397053300946

T. Guhr, Supersymmetry, The Oxford Handbook of Random Matrix Theory (Eds. G. Akemann, J. Baik and P. D. Francesco), Oxford university press, 2015, Chapter 7,135-154. https://doi.org/10.1093/oxfordhb/9780198744191.013.7

L.K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence, RI, 1963. https://doi.org/10.1090/mmono/006

P. Kopel, Linear Statistics of Non-Hermitian Matrices Matching the Real or Complex Ginibre Ensemble to Four Moments, preprint, https://arxiv.org/abs/1510.02987v1.

P. Littelmann, H.-J. Sommers and M.R. Zirnbauer, Superbosonization of invariant random matrix ensembles, Comm. Math. Phys., 283 (2008), 343-395. https://doi.org/10.1007/s00220-008-0535-0

M.L. Mehta, Random matrices and the statistical theory of energy levels, Academic Press, New York-London, 1967.

M.L. Mehta, Random Matrices, Academic Press Inc., Boston, 1991.

A.D. Mirlin and Y. V. Fyodorov, Universality of level correlation function of sparse random matrices, J. Phys. A 24 (1991), 2273-2286. https://doi.org/10.1088/0305-4470/24/10/016

S. O'Rourke and D. Renfrew, Central limit theorem for linear eigenvalue statistics of elliptic random matrices, J. Theoret. Probab. 29 (2016), 1121-1191. https://doi.org/10.1007/s10959-015-0609-9

C. Recher, M. Kieburg, T. Guhr, and M. R. Zirnbauer, Supersymmetry approach to Wishart correlation matrices: Exact results, J. Stat. Phys. 148 (2012), 981-998. https://doi.org/10.1007/s10955-012-0567-x

B. Rider and J. Silverstein, Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. 34 (2006), 2118-2143. https://doi.org/10.1214/009117906000000403

B. Rider and B. Virag, The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN 2 (2007), Art. ID rnm006.

M. Shamis, Density of states for Gaussian unitary ensemble, Gaussian orthogonal ensemble, and interpolating ensembles through supersymmetric approach, J. Math. Phys. 54 (2013), 113505. https://doi.org/10.1063/1.4830013

M. Shcherbina and T. Shcherbina, Transfer matrix approach to 1d random band matrices: density of states, J. Stat. Phys. 164 (2016), 1233-1260. https://doi.org/10.1007/s10955-016-1593-x

M. Shcherbina and T. Shcherbina, Characteristic polynomials for 1D random band matrices from the localization side, Comm. Math. Phys. 351 (2017), 1009-1044. https://doi.org/10.1007/s00220-017-2849-2

M. Shcherbina and T. Shcherbina, Universality for 1d random band matrices: sigma-model approximation, J. Stat. Phys. 172 (2018), 627-664. https://doi.org/10.1007/s10955-018-1969-1

T. Shcherbina, On the correlation function of the characteristic polynomials of the Hermitian Wigner ensemble, Comm. Math. Phys. 308 (2011), 1-21. https://doi.org/10.1007/s00220-011-1316-8

T. Shcherbina, On the correlation functions of the characteristic polynomials of the Hermitian sample covariance matrices, Probab. Theory Related Fields 156 (2013), 449-482. https://doi.org/10.1007/s00440-012-0433-4

E. Strahov and Y.V. Fyodorov, Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach, Comm. Math. Phys. 241 (2003), 343-382. https://doi.org/10.1007/s00220-003-0938-x

T. Tao and V. Vu, Random matrices: universality of ESDs and the circular law, Ann. Probab. 38 (2010), 2023-2065. https://doi.org/10.1214/10-AOP534

T. Tao and V. Vu, Random matrices: universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015), 782-874. https://doi.org/10.1214/13-AOP876

E.B. Vinberg, A Course in Algebra, American Mathematical Society, Providence, RI, 2003. https://doi.org/10.1090/gsm/056

C. Webb and M.D. Wong, On the moments of the characteristic polynomial of a Ginibre random matrix, Proc. Lond. Math. Soc. (3) 118 (2019), 1017-1056. https://doi.org/10.1112/plms.12225

M.R. Zirnbauer, The supersymmetry method of random matrix theory. In: Encyclopedia of mathematical physics, 5, 151-160. Elsevier, 2006. https://doi.org/10.1016/B0-12-512666-2/00068-7

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2022-08-20

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Afanasiev, I. Про кореляцiйнi функцiї характеристичних полiномiв випадкових матриць з незалежними елементами: Iнтерполяцiя мiж комплексним I дiйсним випадками. J. Math. Phys. Anal. Geom. 2022, 18, 159-181.

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