American Institute of Mathematical Sciences

Advanced
  2018, 25: 60-71. doi: 10.3934/era.2018.25.007

A moment method for invariant ensembles

Sho Matsumoto 1, and  Jonathan Novak 2,

1. 

Department of Mathematics, Kagoshima University, Kagoshima, Japan
2. 

Department of Mathematics, University of California, San Diego, USA

Received  April 07, 2018 Revised  September 05, 2018 Published  December 2018

We introduce a new moment method in Random Matrix Theory specifically tailored to the spectral analysis of invariant ensembles. Our method produces a classification of invariant ensembles which exhibit a spectral Law of Large Numbers and yields an explicit description of the limiting eigenvalue distribution when it exists. We discuss the future development and applications of this new moment method.

Keywords: Random matrices, invariant ensembles, moment method.
Mathematics Subject Classification: Primary: 15B52.
Citation: Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007
References:
[1]

A. Borodin, A. Bufetov and G. Olshanski, Limit shapes for growing extreme characters of $U(∞)$, Ann. Appl. Prob., 25 (2015), 2339-2381.  doi: 10.1214/14-AAP1050.  Google Scholar

[2]

A. Bufetov and V. Gorin, Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., 25 (2015), 763-814.  doi: 10.1007/s00039-015-0323-x.  Google Scholar

[3]

H. Cohh, M. Larsen and J. Propp, The shape of a typical boxed plane partition, New York J. Math., 4 (1998), 137-165.   Google Scholar

[4]

B. Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Internat. Math. Res. Not., 17 (2003), 953-982.  doi: 10.1155/S107379280320917X.  Google Scholar

[5]

B. Collins, S. Matsumoto and J. Novak, An Invitation to the Weingarten Calculus, book in preparation. Google Scholar

[6]

B. Collins and P. Śniady, Integration with respect to Haar measure on the unitary, orthogonal and symplectic group, Comm. Math. Phys., 264 (2006), 773-795.  doi: 10.1007/s00220-006-1554-3.  Google Scholar

[7]

B. Conrey, Notes on L-functions and random matrix theory, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,107-162.  Google Scholar

[8]

P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.  Google Scholar

[9]

P. Deift and D. Goev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Lecture Notes in Mathematics, 18, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2009. doi: 10.1090/cln/018.  Google Scholar

[10]

L. Erdos and H.-T. Yau, A Dynamical Approach to Random Matrix Theory Courant Lecture Notes in Mathematics, 28, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2017.  Google Scholar

[11]

P. J. Forrester, Log-Gases and Random Matrices, London Mathematical Society Monographs Series, 34, Princeton University Press, Princeton, NJ, 2010. doi: 10.1515/9781400835416.  Google Scholar

[12]

A. Guionnet and M. Maïda, A Fourier view on the $R$-transform and related asymptotics of spherical integrals, J. Funct. Anal., 222 (2005), 435-490.  doi: 10.1016/j.jfa.2004.09.015.  Google Scholar

[13]

A. Jagannath and T. Trogdon, Random matrices and the New York City subway system, Phys. Rev. E, 96 (2017). doi: 10.1103/PhysRevE.96.030101.  Google Scholar

[14]

K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91 (1998), 151-204.  doi: 10.1215/S0012-7094-98-09108-6.  Google Scholar

[15]

K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys., 209 (2000), 437-476.  doi: 10.1007/s002200050027.  Google Scholar

[16]

R. Kenyon and A. Okounkov, Planar dimers and Harnack curves, Duke Math. J., 131 (2006), 499-524.  doi: 10.1215/S0012-7094-06-13134-4.  Google Scholar

[17]

R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation, Acta Math., 199 (2007), 263-302.  doi: 10.1007/s11511-007-0021-0.  Google Scholar

[18]

R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, Ann. Math. (2), 163 (2006), 1019-1056.  doi: 10.4007/annals.2006.163.1019.  Google Scholar

[19]

S. L. Lauritzen, Thiele: Pioneer in Statistics, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198509721.001.0001.  Google Scholar

[20]

V. A. Marčhenko and L. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sb. NS, 72 (1967), 507-536.   Google Scholar

[21]

S. Matsumoto and J. Novak, in preparation. Google Scholar

[22]

J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Fields Institute Monographs 35, Springer, New York, 2017. doi: 10.1007/978-1-4939-6942-5.  Google Scholar

[23]

A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511735127.  Google Scholar

[24]

J. Novak and P. Śniady, What is... a free cumulant? Notices Amer. Math. Soc., 58 (2011), 300-301.  Google Scholar

[25]

J. Novak, Three lectures on free probability, with illustrations by M. LaCroix, in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Math. Sci. Res. Inst. Publ., 65, Cambridge Univ. Press, New York, 2014,309-383.  Google Scholar

[26]

J. Novak, Lozenge tilings and Hurwitz numbers, J. Stat. Phys., 161 (2015), 509-517.  doi: 10.1007/s10955-015-1330-x.  Google Scholar

[27]

G. Olshanski and A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, in Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2,175, Adv. Math. Sci., 31, Amer. Math. Soc., Providence, RI, 1996,137-175. doi: 10.1090/trans2/175/09.  Google Scholar

[28]

L. Petrov, Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Related Fields, 160 (2014), 429-487.  doi: 10.1007/s00440-013-0532-x.  Google Scholar

[29]

A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys., 207 (1999), 697-733.  doi: 10.1007/s002200050743.  Google Scholar

[30]

D. V. Voiculescu, Limit laws for random matrices and free products, Invent. Math., 104 (1991), 201-220.  doi: 10.1007/BF01245072.  Google Scholar

[31]

D. V. Voiculescu, K. J. Dykema and A. Nica, Free random variables CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[32]

E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2), 62 (1955), 548-564.  doi: 10.2307/1970079.  Google Scholar

[33]

J. Wishart, The generalised product moment distribution in samples from a multivariate normal population, Biometrika, 20A (1928), 32-52.   Google Scholar

show all references

References:
[1]

A. Borodin, A. Bufetov and G. Olshanski, Limit shapes for growing extreme characters of $U(∞)$, Ann. Appl. Prob., 25 (2015), 2339-2381.  doi: 10.1214/14-AAP1050.  Google Scholar

[2]

A. Bufetov and V. Gorin, Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., 25 (2015), 763-814.  doi: 10.1007/s00039-015-0323-x.  Google Scholar

[3]

H. Cohh, M. Larsen and J. Propp, The shape of a typical boxed plane partition, New York J. Math., 4 (1998), 137-165.   Google Scholar

[4]

B. Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Internat. Math. Res. Not., 17 (2003), 953-982.  doi: 10.1155/S107379280320917X.  Google Scholar

[5]

B. Collins, S. Matsumoto and J. Novak, An Invitation to the Weingarten Calculus, book in preparation. Google Scholar

[6]

B. Collins and P. Śniady, Integration with respect to Haar measure on the unitary, orthogonal and symplectic group, Comm. Math. Phys., 264 (2006), 773-795.  doi: 10.1007/s00220-006-1554-3.  Google Scholar

[7]

B. Conrey, Notes on L-functions and random matrix theory, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,107-162.  Google Scholar

[8]

P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.  Google Scholar

[9]

P. Deift and D. Goev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Lecture Notes in Mathematics, 18, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2009. doi: 10.1090/cln/018.  Google Scholar

[10]

L. Erdos and H.-T. Yau, A Dynamical Approach to Random Matrix Theory Courant Lecture Notes in Mathematics, 28, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2017.  Google Scholar

[11]

P. J. Forrester, Log-Gases and Random Matrices, London Mathematical Society Monographs Series, 34, Princeton University Press, Princeton, NJ, 2010. doi: 10.1515/9781400835416.  Google Scholar

[12]

A. Guionnet and M. Maïda, A Fourier view on the $R$-transform and related asymptotics of spherical integrals, J. Funct. Anal., 222 (2005), 435-490.  doi: 10.1016/j.jfa.2004.09.015.  Google Scholar

[13]

A. Jagannath and T. Trogdon, Random matrices and the New York City subway system, Phys. Rev. E, 96 (2017). doi: 10.1103/PhysRevE.96.030101.  Google Scholar

[14]

K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91 (1998), 151-204.  doi: 10.1215/S0012-7094-98-09108-6.  Google Scholar

[15]

K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys., 209 (2000), 437-476.  doi: 10.1007/s002200050027.  Google Scholar

[16]

R. Kenyon and A. Okounkov, Planar dimers and Harnack curves, Duke Math. J., 131 (2006), 499-524.  doi: 10.1215/S0012-7094-06-13134-4.  Google Scholar

[17]

R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation, Acta Math., 199 (2007), 263-302.  doi: 10.1007/s11511-007-0021-0.  Google Scholar

[18]

R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, Ann. Math. (2), 163 (2006), 1019-1056.  doi: 10.4007/annals.2006.163.1019.  Google Scholar

[19]

S. L. Lauritzen, Thiele: Pioneer in Statistics, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198509721.001.0001.  Google Scholar

[20]

V. A. Marčhenko and L. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sb. NS, 72 (1967), 507-536.   Google Scholar

[21]

S. Matsumoto and J. Novak, in preparation. Google Scholar

[22]

J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Fields Institute Monographs 35, Springer, New York, 2017. doi: 10.1007/978-1-4939-6942-5.  Google Scholar

[23]

A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511735127.  Google Scholar

[24]

J. Novak and P. Śniady, What is... a free cumulant? Notices Amer. Math. Soc., 58 (2011), 300-301.  Google Scholar

[25]

J. Novak, Three lectures on free probability, with illustrations by M. LaCroix, in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Math. Sci. Res. Inst. Publ., 65, Cambridge Univ. Press, New York, 2014,309-383.  Google Scholar

[26]

J. Novak, Lozenge tilings and Hurwitz numbers, J. Stat. Phys., 161 (2015), 509-517.  doi: 10.1007/s10955-015-1330-x.  Google Scholar

[27]

G. Olshanski and A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, in Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2,175, Adv. Math. Sci., 31, Amer. Math. Soc., Providence, RI, 1996,137-175. doi: 10.1090/trans2/175/09.  Google Scholar

[28]

L. Petrov, Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Related Fields, 160 (2014), 429-487.  doi: 10.1007/s00440-013-0532-x.  Google Scholar

[29]

A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys., 207 (1999), 697-733.  doi: 10.1007/s002200050743.  Google Scholar

[30]

D. V. Voiculescu, Limit laws for random matrices and free products, Invent. Math., 104 (1991), 201-220.  doi: 10.1007/BF01245072.  Google Scholar

[31]

D. V. Voiculescu, K. J. Dykema and A. Nica, Free random variables CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[32]

E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2), 62 (1955), 548-564.  doi: 10.2307/1970079.  Google Scholar

[33]

J. Wishart, The generalised product moment distribution in samples from a multivariate normal population, Biometrika, 20A (1928), 32-52.   Google Scholar

Figure 1.  A lozenge tiling of a sawtooth domain of rank 6.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83
[2]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[3]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[4]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[5]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[6]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045
[7]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[8]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[9]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055
[10]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[11]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[12]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[13]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[14]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[15]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[16]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263
[17]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[18]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069
[19]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

2019 Impact Factor: 0.5

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences