On the moments of characteristic polynomials

We examine the asymptotics of the moments of characteristic polynomials of N ×N matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as N → ∞. We focus in particular on the Gaussian Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a novel approach to calculate asymptotic formulae for the moments, enabling us to uncover subtle structure not apparent in previous approaches.

[1]  J. Keating.  Recent Perspectives in Random Matrix Theory and Number Theory , 2005 .

[2]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[3]  The Calogero-Sutherland model and polynomials with prescribed symmetry , 1996, solv-int/9609010.

[4]  Persi Diaconis,et al.  Random Matrices, Magic Squares and Matching Polynomials , 2004, Electron. J. Comb..

[5]  Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations , 2002, math-ph/0212050.

[6]  Edouard Brézin,et al.  Characteristic Polynomials of Random Matrices , 2000 .

[7]  Yan V. Fyodorov,et al.  Freezing transitions and extreme values: random matrix theory, and disordered landscapes , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  P. Forrester.  Log-Gases and Random Matrices (LMS-34) , 2010 .

[9]  E. Strahov,et al.  Averages of Characteristic Polynomials in Random Matrix Theory , 2004 .

[10]  Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles , 2020, 2003.08128.

[11]  Yan V. Fyodorov,et al.  An exact formula for general spectral correlation function of random Hermitian matrices , 2002, math-ph/0204051.

[12]  Daniel Bump,et al.  On the Averages of Characteristic Polynomials From Classical Groups , 2005 .

[13]  Hikami,et al.  Characteristic polynomials of random matrices at edge singularities , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[15]  Nina C Snaith,et al.  Random Matrix Theory and L-Functions at s= 1/2 , 2000 .

[16]  J. Keating.  Random matrix theory and number theory , 2009 .

[17]  Alexander Gamburd,et al.  Counting formulas associated with some random matrix averages , 2006, J. Comb. Theory, Ser. A.

[18]  Ida W. Busbridge.  The Evaluation of Certain Integrals Involving Products of Hermite Polynomials , 1939 .

[19]  Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds , 2002, math-ph/0201045.

[20]  N. Simm,et al.  On the distribution of the maximum value of the characteristic polynomial of GUE random matrices , 2015, 1503.07110.

[21]  J. P. Keating,et al.  Integral Moments of L‐Functions , 2002, math/0206018.

[22]  E. Strahov.  Moments of Characteristic Polynomials Enumerate Two-Rowed Lexicographic Arrays , 2003, Electron. J. Comb..

[23]  I. V. Krasovsky.  Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant , 2004 .

[24]  On characteristic polynomials for a generalized chiral random matrix ensemble with a source , 2017, 1711.07061.

[25]  Francesco Mezzadri,et al.  Symmetric function theory and unitary invariant ensembles , 2020, Journal of Mathematical Physics.

[26]  Yan V Fyodorov.  Negative moments of characteristic polynomials of random matrices: Ingham–Siegel integral as an alternative to Hubbard–Stratonovich transformation , 2002 .

[27]  Clusters of near-degenerate levels dominate negative moments of spectral determinants , 2002 .

[28]  Ida W. Busbridge.  Some Integrals Involving Hermite Polynomials , 1948 .

[29]  Y. Fyodorov,et al.  Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function. , 2012, Physical review letters.

[30]  Alexander P. Veselov,et al.  Jacobi-Trudy formula for generalised Schur polynomials , 2009, 0905.2557.

[31]  Yan V. Fyodorov,et al.  Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach , 2002 .

[32]  Microscopic spectrum of the QCD Dirac operator in three dimensions , 2000, hep-th/0009237.

[33]  N. Snaith,et al.  Random Matrix Theory and ζ(1/2+it) , 2000 .

[34]  S. Das.  The freezing transition , 2011 .

[35]  F. Mezzadri,et al.  Rate of convergence of linear functions on the unitary group , 2010, 1009.0695.

[36]  Y. Fyodorov,et al.  Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes , 2015, 1511.04258.

[37]  Jonathan Breuer,et al.  A universality theorem for ratios of random characteristic polynomials , 2012, J. Approx. Theory.

[38]  P. Forrester,et al.  The Calogero-Sutherland Model and Generalized Classical Polynomials , 1996, solv-int/9608004.

[39]  Eugene Strahov,et al.  Products and ratios of characteristic polynomials of random Hermitian matrices , 2003 .

[40]  Universal spectral correlators and massive Dirac operators , 1997, hep-th/9711023.

[41]  Singularity Dominated Strong Fluctuations for Some Random Matrix Averages , 2004, math-ph/0402001.

[42]  High moments of the Riemann zeta-function , 1999, math/9902162.

[43]  Andreev,et al.  Correlators of spectral determinants in quantum chaos. , 1995, Physical review letters.

[44]  P. J. Forrester,et al.  Applications and generalizations of Fisher–Hartwig asymptotics , 2004 .

[45]  J. P. Keating,et al.  Autocorrelation of Random Matrix Polynomials , 2002, math-ph/0208007.

[46]  T. Garoni.  On the asymptotics of some large Hankel determinants generated by Fisher–Hartwig symbols defined on the real line , 2004, math-ph/0411019.