Sum rules via large deviations. (English) Zbl 1327.47028
Summary: In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. R. Killip and B. Simon [Ann. Math. (2) 158, No. 1, 253–321 (2003; Zbl 1050.47025)] have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on \([- 2, 2]\). This sum rule includes a contribution of the atomic part of the measure away from \([- 2, 2]\). In this paper, we recover this sum rule by using probabilistic tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Marchenko-Pastur or Kesten-McKay distributions. As in the semicircular case, these formulas include a contribution of the atomic part appearing away from the support of the reference measure.
MSC:
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |
| 47B80 | Random linear operators |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 60B20 | Random matrices (probabilistic aspects) |
Citations:
Zbl 1050.47025References:
| [1] | Albeverio, S.; Pastur, L.; Shcherbina, M., On the \(1 / n\) expansion for some unitary invariant ensembles of random matrices, Comm. Math. Phys., 224, 1, 271-305 (2001) · Zbl 1038.82039 |
| [2] | Anderson, G.; Guionnet, A.; Zeitouni, O., An Introduction to Random Matrices (2010), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1184.15023 |
| [3] | Auffinger, A.; Ben Arous, G.; Černỳ, J., Random matrices and complexity of spin glasses, Comm. Pure Appl. Math., 66, 2, 165-201 (2013) · Zbl 1269.82066 |
| [4] | Baldi, P., Large deviations and stochastic homogenization, Ann. Mat. Pura Appl., 151, 1, 161-177 (1988) · Zbl 0654.60024 |
| [5] | Ben Arous, G.; Dembo, A.; Guionnet, A., Aging of spherical spin glasses, Probab. Theory Related Fields, 120, 1-67 (2001) · Zbl 0993.60055 |
| [6] | Benaych-Georges, F.; Guionnet, A.; Maïda, M., Large deviations of the extreme eigenvalues of random deformations of matrices, Probab. Theory Related Fields, 154, 3-4, 703-751 (2012) · Zbl 1261.15042 |
| [7] | Borot, G.; Guionnet, A., Asymptotic expansion of \(β\) matrix models in the multi-cut regime (2013), arXiv preprint · Zbl 1344.60012 |
| [8] | Borot, G.; Guionnet, A., Asymptotic expansion of \(β\) matrix models in the one-cut regime, Comm. Math. Phys., 317, 2, 447-483 (2013) · Zbl 1344.60012 |
| [9] | Bourgade, P.; Nikeghbali, A.; Rouault, A., Circular Jacobi ensembles and deformed Verblunsky coefficients, Int. Math. Res. Not., 23, 4357-4394 (2009) · Zbl 1183.33013 |
| [10] | Damanik, D.; Killip, R.; Simon, B., Perturbations of orthogonal polynomials with periodic recursion coefficients, Ann. of Math., 171, 3, 1931-2010 (2010) · Zbl 1194.47031 |
| [11] | Deift, P.; Kriecherbauer, T.; McLaughlin, K.; Venakides, S.; Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math., 52, 11, 1335-1425 (1999) · Zbl 0944.42013 |
| [12] | Dembo, A.; Zeitouni, O., Large Deviations Techniques and Applications (1998), Springer · Zbl 0896.60013 |
| [13] | Dette, H.; Studden, W., The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis, Wiley Series in Probability and Statistics (1997) · Zbl 0886.62002 |
| [14] | Dumitriu, I.; Edelman, A., Matrix models for beta ensembles, J. Math. Phys., 43, 11, 5830-5847 (2002) · Zbl 1060.82020 |
| [15] | Fan, C.; Guionnet, A.; Song, Y.; Wang, A., Convergence of eigenvalues to the support of the limiting measure in critical \(β\) matrix models (2014), arXiv preprint · Zbl 1331.60018 |
| [16] | Gamboa, F.; Lozada-Chang, L.-V., Large deviations for random power moment problem, Ann. Probab., 32, 3B, 2819-2837 (2004) · Zbl 1062.60023 |
| [17] | Gamboa, F.; Nagel, J.; Rouault, A.; Wagener, J., Large deviations for random matricial moment problems, J. Multivariate Anal., 106, 17-35 (2012) · Zbl 1236.15068 |
| [18] | Gamboa, F.; Rouault, A., Canonical moments and random spectral measures, J. Theoret. Probab., 23, 1015-1038 (2010) · Zbl 1204.60009 |
| [19] | Gamboa, F.; Rouault, A., Large deviations for random spectral measures and sum rules, Appl. Math. Res. Express. AMRX, 2, 281-307 (2011) · Zbl 1261.60008 |
| [20] | Hardy, A., A note on large deviations for 2D Coulomb gas with weakly confining potential, Electron. Commun. Probab., 17, 19, 1-12 (2012) · Zbl 1258.60027 |
| [21] | Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91, 1, 151-204 (1998) · Zbl 1039.82504 |
| [22] | Killip, R., Spectral theory via sum rules, (Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math., vol. 76 (2007), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 907-930 · Zbl 1137.47001 |
| [23] | Killip, R.; Nenciu, I., Matrix models for circular ensembles, Int. Math. Res. Not., 50, 2665-2701 (2004) · Zbl 1255.82004 |
| [24] | Killip, R.; Simon, B., Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math., 158, 1, 253-321 (2003) · Zbl 1050.47025 |
| [25] | Krishnapur, M.; Rider, B.; Virag, B., Universality of the stochastic Airy operator (2013) |
| [26] | Kupin, S., Spectral properties of Jacobi matrices and sum rules of special form, J. Funct. Anal., 227, 1, 1-29 (2005) · Zbl 1083.47025 |
| [27] | Nazarov, F.; Peherstorfer, F.; Volberg, A.; Yuditskii, P., On generalized sum rules for Jacobi matrices, Int. Math. Res. Not., 3, 155-186 (2005) · Zbl 1089.47025 |
| [28] | Rockafellar, R., Integrals which are convex functionals, II, Pacific J. Math., 39, 2, 439-469 (1971) · Zbl 0236.46031 |
| [29] | Saitoh, N.; Yoshida, H., The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory, Probab. Math. Statist., 21, 1, 159-170 (2001), Acta Univ. Wratislav. No. 2298 · Zbl 1020.46019 |
| [30] | Serfaty, S., Coulomb gases and Ginzburg-Landau vortices (2014) · Zbl 1291.82142 |
| [31] | Simon, B., Orthogonal polynomials with exponentially decaying recursion coefficients, (Probability and Mathematical Physics (2007)), 453-463 · Zbl 1143.42306 |
| [32] | Simon, B., Szegő’s Theorem and Its Descendants, M. B. Porter Lectures (2011), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1230.33001 |
| [33] | Trotter, H., Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő, Adv. Math., 54, 1, 67-82 (1984) · Zbl 0562.15005 |
| [34] | Xia, N.; Qin, Y.; Bai, Z., Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix, Ann. Statist., 41, 5, 2572-2607 (2013) · Zbl 1285.15018 |
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