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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

S. Matsumoto and J. Novak, in preparation.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

S. Matsumoto and J. Novak, in preparation.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

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