2014

Universality of mesoscopic fluctuations for orthogonal polynomial ensembles

Breuer, J., Duits, M.

Understand

We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures).

The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical.
Our main tool is an analysis of the Green's function for the associated Jacobi matrices.
As a particular consequence we obtain a Central Limit Theorem for the modified Jacobi Unitary Ensembles on all mesosopic scales.

Built on

M. Reed and B. Simon, Methods of Modern Mathematical Physics,
1972
Earlier work this paper cites.
J. M. Combes and L. Thomas, Asymptotic behavior of eigenfunctions for multi-particle Schrödinger operators
1973
Earlier work this paper cites.
K. Case and J. Geronimo, Scattering theory and polynomials orthogonal on the unit circle , J. Math. Phys. 20
1979
Earlier work this paper cites.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, III: Scattering Theory
1979
Earlier work this paper cites.
E. A. Rahmanov, On Steklov’s conjecture in the theory of orthogonal polynomials
1980
Earlier work this paper cites.
D. J. Gilbert, D. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators
1987
Earlier work this paper cites.
O. Costin and J. L. Lebowitz, Gaussian fluctuation in random matrices , Phys. Rev. Lett. 75
1995
Earlier work this paper cites.
B. Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators
1996
Earlier work this paper cites.
A. Boutet de Monvel and A. Khorunzy, Asymptotic distribution of smoothed eigenvalue density I. Gaussian random matrices, Random Oper. Stochastic Equations 7
1999
Earlier work this paper cites.
A. Boutet de Monvel and A. Khorunzy, Asymptotic distribution of smoothed eigenvalue density II. Wigner random matrices. Random Oper. Stochastic Equations 7
1999
Earlier work this paper cites.
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , Courant Institute Lecture Notes, 3
1999
Earlier work this paper cites.
P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schrodinger operators with square summable potentials , Commun. Math. Phys. 203
1999
Earlier work this paper cites.

Similar

S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra, I. Half-line operators
1999
Cited alongside, same era.
E. Basor and H. Widom, On a Toeplitz determinant identity of Borodin and Okounkov , Integral Equations and Operator Theory 37 (2000), no. 4, 397–401
2000
Cited alongside, same era.
A. Borodin and A. Okounkov, A Fredholm determinant formula for Toeplitz determinants , Integral Equations Operator Theory 37
2000
Cited alongside, same era.
A. Soshnikov, The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28
2000
Cited alongside, same era.
A. Soshnikov, Determinantal random point fields , Uspekhi Mat. Nauk 55
2000
Cited alongside, same era.
A. B. J. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble , Int. Math. Res. Not. 2002
2002
Cited alongside, same era.
P. Diaconis, Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.§. ) 40
2003
Cited alongside, same era.
T. A. Ehrhardt, Generalization of Pincus’ formula and Toeplitz operator determinants Arch. Math. 80
2003
Cited alongside, same era.
A. Kiselev, Y. Last and B. Simon, Stability of singular spectral types under decaying perturbations
2003
Cited alongside, same era.
R. Lyons, Determinantal probability measures , Publ. Math. Inst. Hautes Etudes Sci. 98
2003
Cited alongside, same era.
J. Korevaar, Tauberian Theory, A century of developments, Grundlehren der Mathematischen Wissenschaften 329, 2004. Springer-Verlag Berlin Heidelberg
2004
Cited alongside, same era.
A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] , Adv. Math. 188
2004
Cited alongside, same era.

Then

W. König, Orthogonal polynomial ensembles in probability theory , Probab. Surveys 2
2005
Later among the works it cites.
B. Simon, Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005. viii+150
2005
Later among the works it cites.
K. Johansson, Random matrices and determinantal processes , Mathematical Statistical Physics, Elsevier B.V. Amsterdam (2006) 1–55
2006
Later among the works it cites.
L. Pastur, Limiting laws of linear eigenvalue statistics for Hermitian matrix models , J. Math. Phys. 47
2006
Later among the works it cites.
G. W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices , Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010
2010
Later among the works it cites.
B. Simon, Szegő’s Theorem and its Descendants: Spectral Theory for L 2 L^{2} Perturbations of Orthogonal Polynomials , M. B. Porter Lecture Series, Princeton Univ. Press, Princeton, NJ, 2011
2011
Later among the works it cites.
The Oxford handbook of random matrix theory , Edited by G. Akemann, J. Baik and P. Di Francesco. Oxford University Press, Oxford, 2011
Original
2011
Later among the works it cites.
M. Shcherbina, Fluctuations of linear eigenvalue statistics of matrix models in the multi-cut regime , J. Stat. Phys. 151
2013
Later among the works it cites.
J. Breuer and M. Duits, Nevai 265 The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles , Adv. Math. 265
2014
Closest in time.
J. Breuer, Y. Last and B. Simon, Stability of Asymptotics of Christoffel-Darboux Kernels , Commun. Math. Phys. (2014), 330
2014
Closest in time.
R. Killip, Perturbations of 1-dimensional Schrödinger operators preserving the absolutely continuous spectrum
2061
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…