×

Universality in several-matrix models via approximate transport maps. (English) Zbl 1402.60009

Many recent papers have paid attention to the question of universality of the distribution of the local statistics of Wigner matrices. The aim of this sizeable article is to provide new universality results for general perturbative several-matrix models. The authors prove that if \(X_1^N,\ldots,X_d^N\) are independent \(N\times N\) matrices in the Gaussian Unitary Ensemble (GUE), if \(B_1^N,\ldots, B_m^N\) are Hermitian deterministic matrices and if \(p\) is a polynomial in \(d+m\) variables which is a perturbation of \(x_1\), then the eigenvalues of \(p(X_1^N,\ldots,X_d^N, B_1^N,\ldots,B_m^N)\) fluctuate as the eigenvalues of \(X_1^N\). In particular, when \(p\) depends only on \(d\) variables and \(p(x_1,\ldots,x_d)=x_1+ \varepsilon Q((x_1,\ldots,x_d)\) with \(Q\) self-adjoint, then the properly normalized fluctuations of the eigenvalues of \(p(X_1^N,\ldots,X_d^N)\) follow the sine-kernel distribution inside the bulk and the Tracy-Widom law at he edges of the support of the semi-circle law. The result derives from an universality property for unitarily invariant matrices interacting via a potential, which is obtained by finding a map from the law of the eigenvalues of independent GUE or GOE matrices to a probability measure that approximates the matrix model. As it is necessary to treat quadratic statistics, a large \(N\)-expansion for integrals over the unitary and orthogonal group is included in the proof.

MSC:

60B20 Random matrices (probabilistic aspects)
49Q20 Variational problems in a geometric measure-theoretic setting
15B52 Random matrices (algebraic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Akemann, G.; Ipsen, J. R., Recent exact and asymptotic results for products of independent random matrices, Acta Phys. Polon. B, 46, 1747-1784, (2015) · Zbl 1371.60008
[2] Anderson, G. W., Guionnet, A. & Zeitouni, O., An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge Univ. Press, Cambridge, 2010. · Zbl 1184.15023
[3] Aptekarev, A. I.; Bleher, P. M.; Kuijlaars, A. B. J., Large \(n\) limit of Gaussian random matrices with external source. II, Comm. Math. Phys., 259, 367-389, (2005) · Zbl 1129.82014
[4] Bekerman, F., Transport maps for \({β}\)-matrix models in the multi-cut regime. Preprint, 2015. arXiv:1512.00302 [math.PR]. · Zbl 1390.60028
[5] Bekerman, F.; Figalli, A.; Guionnet, A., Transport maps for \({β}\)-matrix models and universality, Comm. Math. Phys., 338, 589-619, (2015) · Zbl 1330.49046
[6] Ben Arous, G.; Bourgade, P., Extreme gaps between eigenvalues of random matrices, Ann. Probab., 41, 2648-2681, (2013) · Zbl 1282.60008
[7] Ben Arous, G.; Guionnet, A., Large deviations for wigner’s law and voiculescu’s non-commutative entropy, Probab. Theory Related Fields, 108, 517-542, (1997) · Zbl 0954.60029
[8] Bertola, M., Two-matrix models and biorthogonal polynomials, in The Oxford Handbook of Random Matrix Theory, pp. 310-328. Oxford Univ. Press, Oxford, 2011. · Zbl 1237.81111
[9] Borot, G.; Guionnet, A., Asymptotic expansion of \({β}\) matrix models in the one-cut regime, Comm. Math. Phys., 317, 447-483, (2013) · Zbl 1344.60012
[10] Borot, G. & Guionnet, A., Asymptotic expansion of beta matrix models in the multi-cut regime. Preprint, 2013. arXiv:1303.1045 [math.PR]. · Zbl 1344.60012
[11] Borot, G.; Guionnet, A.; Kozlowski, K. K., Large-\({N}\) asymptotic expansion for Mean field models with Coulomb gas interaction, Int. Math. Res. Not. IMRN, 2015, 10451-10524, (2015) · Zbl 1332.82069
[12] Bourgade, P., Erdős, L. & Yau, H.-T., Bulk universality of general \({β}\)-ensembles with non-convex potential. J. Math. Phys., 53 (2012), 095221, 19 pp. · Zbl 1278.82032
[13] Bourgade, P.; Erdős, L.; Yau, H.-T., Edge universality of beta ensembles, Comm. Math. Phys., 332, 261-353, (2014) · Zbl 1306.82010
[14] Bourgade, P.; Erdős, L.; Yau, H.-T., Universality of general \({β}\)-ensembles, Duke Math. J., 163, 1127-1190, (2014) · Zbl 1298.15040
[15] Bourgade, P.; Erdős, L.; Yau, H.-T.; Yin, J., Fixed energy universality for generalized Wigner matrices, Comm. Pure Appl. Math., 69, 1815-1881, (2016) · Zbl 1354.15025
[16] Brézin, E.; Itzykson, C.; Parisi, G.; Zuber, J. B., Planar diagrams, Comm. Math. Phys., 59, 35-51, (1978) · Zbl 0997.81548
[17] Capitaine, M.; Péché, S., Fluctuations at the edges of the spectrum of the full rank deformed GUE, Probab. Theory Related Fields, 165, 117-161, (2016) · Zbl 1342.15029
[18] Collins, B.; Guionnet, A.; Maurel-Segala, É., Asymptotics of unitary and orthogonal matrix integrals, Adv. Math., 222, 172-215, (2009) · Zbl 1184.15024
[19] Deift, P. A., Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, 3. Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1999. · Zbl 0997.47033
[20] Deift, P. A.; Gioev, D., Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices, Comm. Pure Appl. Math., 60, 867-910, (2007) · Zbl 1119.15022
[21] Deift, P. A. & Gioev, D., Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Pap. IMRP, 2 (2007), Art. ID rpm004, 116 pp. · Zbl 1136.82021
[22] Deift, P. A. & Gioev, D., Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics, 18. Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 2009. · Zbl 1171.15023
[23] Erdős, L., Universality of Wigner random matrices, in XVI International Congress on Mathematical Physics, pp. 86-105. World Sci., Hackensack, NJ, 2010. · Zbl 1205.82080
[24] Erdős, L.; Knowles, A.; Yau, H.-T.; Yin, J., Delocalization and diffusion profile for random band matrices, Comm. Math. Phys., 323, 367-416, (2013) · Zbl 1279.15027
[25] Erdős, L.; Péché, S.; Ramírez J., A.; Schlein, B.; Yau, H.-T., Bulk universality for Wigner matrices, Comm. Pure Appl. Math., 63, 895-925, (2010) · Zbl 1216.15025
[26] Erdős, L.; Schlein, B.; Yau, H.-T.; Yin, J., The local relaxation flow approach to universality of the local statistics for random matrices, Ann. Inst. Henri Poincaré Probab. Stat., 48, 1-46, (2012) · Zbl 1285.82029
[27] Erdős, L.; Yau, H.-T., Universality of local spectral statistics of random matrices, Bull. Amer. Math. Soc., 49, 377-414, (2012) · Zbl 1263.15032
[28] Erdős, L.; Yau, H.-T., Gap universality of generalized Wigner and \({β}\)-ensembles, J. Eur. Math. Soc. (JEMS), 17, 1927-2036, (2015) · Zbl 1333.15031
[29] Erdős, L.; Yau, H.-T.; Yin, J., Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math., 229, 1435-1515, (2012) · Zbl 1238.15017
[30] Eynard, B.; Bonnet, G., The Potts-\(q\) random matrix model: loop equations, critical exponents, and rational case, Phys. Lett. B, 463, 273-279, (1999) · Zbl 1037.82522
[31] Forrester, P. J., Log-Gases and Random Matrices. London Mathematical Society Monographs Series, 34. Princeton Univ. Press, Princeton, NJ, 2010. · Zbl 1217.82003
[32] Götze, F.; Venker, M., Local universality of repulsive particle systems and random matrices, Ann. Probab., 42, 2207-2242, (2014) · Zbl 1301.60009
[33] Guionnet, A., Jones, V. F. R. & Shlyakhtenko, D., Random matrices, free probability, planar algebras and subfactors, in Quanta of Maths, Clay Math. Proc., 11, pp. 201-239. Amer. Math. Soc., Providence, RI, 2010. · Zbl 1219.46057
[34] Guionnet, A.; Jones, V. F. R.; Shlyakhtenko, D.; Zinn-Justin, P., Loop models, random matrices and planar algebras, Comm. Math. Phys., 316, 45-97, (2012) · Zbl 1277.82013
[35] Guionnet, A.; Maurel-Segala, É., Combinatorial aspects of matrix models, ALEA Lat. Am. J. Probab. Math. Stat., 1, 241-279, (2006) · Zbl 1110.15021
[36] Guionnet, A.; Maurel-Segala, É., Second order asymptotics for matrix models, Ann. Probab., 35, 2160-2212, (2007) · Zbl 1129.15020
[37] Guionnet, A.; Novak, J., Asymptotics of unitary multimatrix models: the Schwinger-Dyson lattice and topological recursion, J. Funct. Anal., 268, 851-2905, (2015) · Zbl 1335.46057
[38] Guionnet, A.; Shlyakhtenko, D., Free monotone transport, Invent. Math., 197, 613-661, (2014) · Zbl 1312.46059
[39] Haagerup, U.; Thorbjørnsen, S., A new application of random matrices: \({{\rm Ext}(C^*_{\rm red}(F_2))}\)is not a group, Ann. of Math., 162, 711-775, (2005) · Zbl 1103.46032
[40] Kostov, I., Two-dimensional quantum gravity, in The Oxford Handbook of Random Matrix Theory, pp. 619-640. Oxford Univ. Press, Oxford, 2011. · Zbl 1236.81166
[41] Kriecherbauer, T. & Shcherbina, M., Fluctuations of eigenvalues of matrix models and their applications. Preprint, 2010. arXiv:1003.6121 [math-ph]. · Zbl 1263.15032
[42] Kriecherbauer, T. & Venker, M., Edge statistics for a class of repulsive particle systems. Preprint, 2015. arXiv:1501.07501 [math.PR]. · Zbl 1456.60022
[43] Krishnapur, M.; Rider, B.; Virág, B., Universality of the stochastic Airy operator, Comm. Pure Appl. Math., 69, 145-199, (2016) · Zbl 1364.60015
[44] Lee, J. O.; Schnelli, K.; Stetler, B.; Yau, H.-T., Bulk universality for deformed Wigner matrices, Ann. Probab., 44, 2349-2425, (2016) · Zbl 1346.15037
[45] Levin, E.; Lubinsky, D. S., Universality limits in the bulk for varying measures, Adv. Math., 219, 743-779, (2008) · Zbl 1176.28014
[46] Liu, D. Z.; Wang, Y., Universality for products of random matrices I: Ginibre and truncated unitary cases, Int. Math. Res. Not. IMRN, 2016, 3473-3524, (2016) · Zbl 1404.60016
[47] Lubinsky, D. S., Universality limits via “old style” analysis, in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Math. Sci. Res. Inst. Publ., 65, pp. 277-292. Cambridge Univ. Press, New York, 2014. · Zbl 1326.15060
[48] Maïda, M.; Maurel-Segala, É., Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices, Probab. Theory Related Fields, 159, 329-356, (2014) · Zbl 1305.46057
[49] Male, C., The norm of polynomials in large random and deterministic matrices, Probab. Theory Related Fields, 154, 477-532, (2012) · Zbl 1269.15039
[50] Maurel-Segala, É., High order expansion of matrix models and enumeration of maps. Preprint, 2006. arXiv:math/0608192 [math.PR]. · Zbl 1110.15021
[51] Mehta, M. L., A method of integration over matrix variables, Comm. Math. Phys., 79, 327-340, (1981) · Zbl 0471.28007
[52] Mehta, M. L., Random Matrices. Pure and Applied Mathematics, 142. Elsevier/Academic Press, Amsterdam, 2004. · Zbl 0789.35152
[53] Nelson, B., Free transport for finite depth subfactor planar algebras, J. Funct. Anal., 268, 2586-2620, (2015) · Zbl 1406.46051
[54] Ramírez, J. A.; Rider, B.; Virág, B., Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc., 24, 919-944, (2011) · Zbl 1239.60005
[55] Shcherbina, M., On universality for orthogonal ensembles of random matrices, Comm. Math. Phys., 285, 957-974, (2009) · Zbl 1190.82020
[56] Shcherbina, M., Change of variables as a method to study general \({β}\)-models: bulk universality. J. Math. Phys., 55 (2014), 043504, 23 pp. · Zbl 1296.82032
[57] Shcherbina, T., Universality of the local regime for the block band matrices with a finite number of blocks, J. Stat. Phys., 155, 466-499, (2014) · Zbl 1305.82032
[58] Tao, T., The asymptotic distribution of a single eigenvalue gap of a Wigner matrix, Probab. Theory Related Fields, 157, 81-106, (2013) · Zbl 1280.15023
[59] Tao, T.; Vu, V., Random matrices: universality of ESDs and the circular law, Ann. Probab., 38, 2023-2065, (2010) · Zbl 1203.15025
[60] Tao, T.; Vu, V., Random matrices: universality of local eigenvalue statistics, Acta Math., 206, 127-204, (2011) · Zbl 1217.15043
[61] Tao, T.; Vu, V., Random covariance matrices: universality of local statistics of eigenvalues, Ann. Probab., 40, 1285-1315, (2012) · Zbl 1247.15036
[62] Tao, T.; Vu, V., Random matrices: universality of local spectral statistics of non-Hermitian matrices, Ann. Probab., 43, 782-874, (2015) · Zbl 1316.15042
[63] Tracy, C. A.; Widom, H., Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159, 151-174, (1994) · Zbl 0789.35152
[64] Tracy, C. A.; Widom, H., Level spacing distributions and the Bessel kernel, Comm. Math. Phys., 161, 289-309, (1994) · Zbl 0808.35145
[65] Valkó, B.; Virág, B., Continuum limits of random matrices and the Brownian carousel, Invent. Math., 177, 463-508, (2009) · Zbl 1204.60012
[66] Venker, M., Particle systems with repulsion exponent \({β}\) and random matrices, Electron. Commun. Probab., 18, 12, (2013) · Zbl 1297.15040
[67] Voiculescu, D., Limit laws for random matrices and free products, Invent. Math., 104, 201-220, (1991) · Zbl 0736.60007
[68] Wigner, E. P., Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math., 62, 548-564, (1955) · Zbl 0067.08403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.