Rectangular random matrices, related convolution. (English) Zbl 1171.15022
The paper deals with the modeling of asymptotic collective behavior of independent rectangular random matrices. Within such modeling it is possible to compute the limit of the asymptotic normalized traces and thus to give the asymptotic normalized trace of any noncommutative polynomial in these independent random matrices. Since the normalized trace of the \(k\)th power of a matrix is the \(k\)th moment of its spectral law it is possible to give the asymptotic singular law of any polynomial of these random matrices.
It is proposed a way to compute the asymptotic normalized trace of products of random matrices taken among a family of independent rectangular random matrices, whose sizes tend to infinity, but with different rates. The modeling of asymptotics of rectangular random matrices enables to define a binary operation for \(\lambda \in [0,1]\) on the set of symmetric probability measures, called free convolution with ratio \(\lambda\). This binary operation allows to determine singular values of a sum of two independent rectangular random matrices, whose dimensions tend to infinity, from their singular laws. Finally, examples of rectangular \(R\)-transforms of symmetric probability measures are given together with examples of computation of rectangular free convolutions.
It is proposed a way to compute the asymptotic normalized trace of products of random matrices taken among a family of independent rectangular random matrices, whose sizes tend to infinity, but with different rates. The modeling of asymptotics of rectangular random matrices enables to define a binary operation for \(\lambda \in [0,1]\) on the set of symmetric probability measures, called free convolution with ratio \(\lambda\). This binary operation allows to determine singular values of a sum of two independent rectangular random matrices, whose dimensions tend to infinity, from their singular laws. Finally, examples of rectangular \(R\)-transforms of symmetric probability measures are given together with examples of computation of rectangular free convolutions.
Reviewer: Václav Burjan (Praha)
References:
| [1] | Akhiezer, N.I.: The classical moment problem. Moscou (1961) · Zbl 0124.06202 |
| [2] | Belinschi, S.T.: The Lebesgue decomposition of the free additive convolution of two probability distributions, preprint (2006), ArXiv math.OA/0603104 · Zbl 1390.46059 |
| [3] | Belinschi, S.T., Benaych-Georges, F., Guionnet, A.: Regularization by free additive convolution, square and rectangular cases, preprint (2007), ArXiv · Zbl 1187.46055 |
| [4] | Benaych-Georges, F.: Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation. Probability Theory and Related Fields, vol. 139, nos. 1-2/September 2007, pp. 143-189 · Zbl 1129.15019 |
| [5] | Benaych-Georges, F.: Rectangular random matrices, related free entropy and free Fisher’s information. J. Oper. Theory (2008, in press) · Zbl 1224.46121 |
| [6] | Bercovici, H.; Pata, V., with an appendix by Biane, P. Stable laws and domains of attraction in free probability theory, Ann. Math., 149, 1023-1060 (1999) · Zbl 0945.46046 · doi:10.2307/121080 |
| [7] | Bercovici, H.; Voiculescu, D., Free convolution of measures with unbounded supports, Indiana Univ. Math. J., 42, 733-773 (1993) · Zbl 0806.46070 · doi:10.1512/iumj.1993.42.42033 |
| [8] | Billingsley, P., Convergence of Probability Measures (1968), London: Wiley, London · Zbl 0172.21201 |
| [9] | Capitaine, M.; Casalis, M., Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices, Indiana Univ. Math. J., 53, 2, 397-431 (2004) · Zbl 1063.46054 · doi:10.1512/iumj.2004.53.2325 |
| [10] | Capitaine, M., Donati-Martin, C.: Strong asymptotic freeness for Wigner and Wishart matrices, preprint (2005) · Zbl 1162.15013 |
| [11] | Donoghue, W., Monotone Matrix Functions and Analytic Continuation (1974), New York: Springer, New York · Zbl 0278.30004 |
| [12] | Dozier, B., Silverstein, J.: Analysis of the limiting distribution of large dimensional information-plus-noise-type matrices, preprint (2004) · Zbl 1127.62015 |
| [13] | Guionnet, A.; Zeitouni, O., Concentration of the spectral measure for large matrices, Electron. Comm. Probab., 5, 119-136 (2000) · Zbl 0969.15010 |
| [14] | Haagerup, U.; Larsen, F., Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras, J. Funct. Anal., 176, 331-367 (2000) · Zbl 0984.46042 · doi:10.1006/jfan.2000.3610 |
| [15] | Hachem, W., Loubaton, P., Najim, J.: The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile, preprint (2004) · Zbl 1109.15021 |
| [16] | Hachem, W., Loubaton, P., Najim, J.: Deterministic equivalents for certain functionals of large random matrices, preprint, July (2005) · Zbl 1181.15043 |
| [17] | Hiai, F., Petz, D.: The semicircle law, free random variables, and entropy. Am. Math. Soc., Mathematical Surveys and Monographs, vol. 77 (2000) · Zbl 0955.46037 |
| [18] | Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0729.15001 |
| [19] | Mehta, M. L., Random Matrices and the Statistical Theory of Energy Levels (1967), New York: Academic Press, New York · Zbl 0925.60011 |
| [20] | Nelson, E., Notes on non-commutative integration, J. Funct. Anal., 15, 103-116 (1974) · Zbl 0292.46030 · doi:10.1016/0022-1236(74)90014-7 |
| [21] | Nica, A.; Shlyakhtenko, D., Speicher, Roland Some minimization problems for the free analogue of the Fisher information, Adv. Math., 141, 2, 282-321 (1999) · Zbl 0929.46052 · doi:10.1006/aima.1998.1780 |
| [22] | Nica, A.; Shlyakhtenko, D., Speicher, Roland Operator-valued distributions. I. Characterizations of freeness, Int. Math. Res. Notes, 29, 1509-1538 (2002) · Zbl 1007.46052 · doi:10.1155/S1073792802201038 |
| [23] | Pastur, L., Lejay, A.: Matrices aléatoires: statistique asymptotique des valeurs propres. Séminaire de Probabilités XXXVI, Lectures Notes in Mathematics, vol. 1801, Springer, Heidelberg (2002) |
| [24] | Rota, Gian-Carlo On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 2, pp. 340-368 (1964) · Zbl 0121.02406 |
| [25] | Shlyakhtenko, D., Random Gaussian band matrices and freeness with amalgamation, Int. Math. Res. Notices, 20, 1013-1025 (1996) · Zbl 0872.15018 · doi:10.1155/S1073792896000633 |
| [26] | Speicher, R., Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann., 298, 611-628 (1994) · Zbl 0791.06010 · doi:10.1007/BF01459754 |
| [27] | Speicher, R., Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Am. Math. Soc., 132, 627 (1998) · Zbl 0935.46056 |
| [28] | Speicher, R.: Notes of my lectures on Combinatorics of Free Probability (IHP, Paris, 1999). Available on http://www.mast.queensu.ca/ speicher (1999) |
| [29] | Śniady, P.; Speicher, R., Continuous family of invariant subspaces for R-diagonal operators, Invent. Math., 146, 329-363 (2001) · Zbl 1032.46077 · doi:10.1007/s002220100166 |
| [30] | Voiculescu, D., Limit laws for random matrices and free products, Invent. Math., 104, 1, 201-220 (1991) · Zbl 0736.60007 · doi:10.1007/BF01245072 |
| [31] | Voiculescu, D., Operations on certain non-commutative operator-valued random variables. Recent advances in operator algebras (Orléans, 1992), AstéRisque, 232, 243-275 (1995) · Zbl 0839.46060 |
| [32] | Voiculescu, D., A strengthened asymptotic freeness result for random matrices with applications to free entropy, Int. Math. Res. Notices, 1, 41-63 (1998) · Zbl 0895.60004 · doi:10.1155/S107379289800004X |
| [33] | Voiculescu, D.V., Dykema, K., Nica, A.: Free random variables. CRM Monograghs Series No. 1, Am. Math. Soc., Providence, RI (1992) · Zbl 0795.46049 |
| [34] | Wigner, E., On the distribution of the roots of certain symmetric matrices, Ann. Math., 67, 325-327 (1958) · Zbl 0085.13203 · doi:10.2307/1970008 |
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