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

A moment method for invariant ensembles

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.

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. BorodinA. 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. CohhM. 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. KenyonA. 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. BorodinA. 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. CohhM. 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. KenyonA. 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.
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