Applications of realizations (aka linearizations) to free probability. (English) Zbl 1376.81026
Summary: We show how the combination of new “linearization” ideas in free probability theory with the powerful “realization” machinery – developed over the last 50 years in fields including systems engineering and automata theory – allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type \(\mathrm{II}_1\) von Neumann algebras, with a precise control of the domains of the rational expressions. The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.
MSC:
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
| 47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
| 60B20 | Random matrices (probabilistic aspects) |
| 46L10 | General theory of von Neumann algebras |
Keywords:
free probability; non-commutative rational functions; descriptor realizations; linearizationsReferences:
| [1] | Amitsur, S. A., Rational identities and applications to algebra and geometry, J. Algebra, 3, 3, 304-359 (May 1966) · Zbl 0203.04003 |
| [2] | Anderson, G. W., Convergence of the largest singular value of a polynomial in independent Wigner matrices, Ann. Probab., 41, 2103-2181 (2013) · Zbl 1282.60007 |
| [3] | Ball, J. A.; Malakorn, T.; Groenewald, G., Structured noncommutative multidimensional linear systems, SIAM J. Control Optim., 44, 4, 1474-1528 (2005) · Zbl 1139.93006 |
| [4] | Beck, C. L., On formal power series representations of uncertain systems, IEEE Trans. Automat. Control, 46, 2, 314-319 (2001) · Zbl 0992.93009 |
| [5] | Belinschi, S. T.; Popa, M.; Vinnikov, V., Infinite divisibility and a non-commutative Boolean-to-free Bercovici-Pata bijection, J. Funct. Anal., 262, 1, 94-123 (2012) · Zbl 1247.46054 |
| [6] | Belinschi, S.; Mai, T.; Speicher, R., Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem, J. Reine Angew. Math. (2013) |
| [7] | Belinschi, S. T.; Sniady, P.; Speicher, R., Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method (2015), preprint · Zbl 1376.15025 |
| [8] | Berstel, J.; Reutenauer, C., Rational Series and Their Languages, EATCS Monographs on Theoretical Computer Science (1984), Springer · Zbl 0573.68037 |
| [9] | Bochnak, J.; Coste, M.; Roy, M., Real Algebraic Geometry, 1-430 (1998) · Zbl 0912.14023 |
| [10] | Brown, L. G., Lidskii’s theorem in the type II case, (Geometric Methods in Operator Algebras. Geometric Methods in Operator Algebras, Proc. US-Jap. Semin., Kyoto/Jap., 1983. Geometric Methods in Operator Algebras. Geometric Methods in Operator Algebras, Proc. US-Jap. Semin., Kyoto/Jap., 1983, Pitman Res. Notes Math. Ser., vol. 123 (1986)), 1-35 · Zbl 0646.46058 |
| [11] | Cohn, P. M., Free Rings and Their Relations (1971), Academic Press: Academic Press London · Zbl 0232.16003 |
| [12] | Cohn, P. M., Free Ideal Rings and Localization in General Rings, 596 (2006), Cambridge University Press · Zbl 1114.16001 |
| [13] | Cohn, P. M.; Reutenauer, C., On the construction of the free field, Internat. J. Algebra Comput., 9.03n04, 307-323 (1999) · Zbl 1040.16015 |
| [14] | Fliess, M., Matrices de Hankel, J. Math. Pures Appl.. J. Math. Pures Appl., J. Math. Pures Appl., 54, 197-222 (1975), (Erratum) · Zbl 0315.94051 |
| [15] | Fliess, M., Sur Divers Produits Series Formalles, Bull. Soc. Math. France, 102, 184-191 (1974) · Zbl 0313.13021 |
| [16] | 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 |
| [17] | Haagerup, U.; Thorbjørnsen, S., A new application of random matrices: \(Ext(C_{red}^\ast(F_2))\) is not a group, Ann. of Math., 162 (2005) · Zbl 1103.46032 |
| [18] | Haagerup, U.; Schultz, H.; Thorbjørnsen, S., A random matrix approach to the lack of projections in \(C_{red}^\ast(F_2)\), Adv. Math., 204, 1-83 (2006) · Zbl 1109.15020 |
| [19] | Helton, J. W.; McCullough, S. A.; Vinnikov, V., Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal., 240, 1, 105-191 (2006) · Zbl 1135.47005 |
| [20] | Hiai, F.; Petz, D., The Semicircle Law, Free Random Variables and Entropy (2000), American Mathematical Society · Zbl 0955.46037 |
| [21] | Janik, R. A.; Nowak, M. A.; Papp, G.; Zahed, I., Non-hermitian random matrix models, Nuclear Phys. B, 501, 3, 603-642 (1997) · Zbl 0933.82023 |
| [22] | Kaliuzhnyi-Verbovetskyi, D. S.; Vinnikov, V., Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting, Linear Algebra Appl., 430, 869-889 (2009) · Zbl 1217.47032 |
| [23] | Kaliuzhnyi-Verbovetskyi, D. S.; Vinnikov, V., Noncommutative rational functions, their difference-differential calculus and realizations, Multidimens. Syst. Signal Process., 23, 1-2, 49-77 (2012) · Zbl 1255.93073 |
| [24] | Kaliuzhnyi-Verbovetskyi, D. S.; Vinnikov, V., Foundations of Free Noncommutative Function Theory, Mathematical Surveys and Monographs, vol. 199 (2012), American Mathematical Society: American Mathematical Society Providence, RI, preprint |
| [25] | Kalman, R. E., Mathematical description of linear dynamical systems, SIAM J. of Control, Ser. A, 1, 2, 152-192 (1963) · Zbl 0145.34301 |
| [26] | Kleene, S. C., Representation of events in nerve nets and finite automata, (Automata Studies (1956), Princeton University Press: Princeton University Press Princeton, NJ, USA), 3-41 |
| [27] | Linnell, P. A., Noncommutative localization in group rings, (Noncommutative Localization in Algebra and Topology. Noncommutative Localization in Algebra and Topology, London Math. Soc. Lecture Note Ser., vol. 330 (2006), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 1123.16016 |
| [28] | Malcolmson, P., A prime matrix ideal yields a skew field, J. Lond. Math. Soc., 18, 221-233 (1978) · Zbl 0406.16013 |
| [29] | Mingo, J. A.; Speicher, R., Free Probability and Random Matrices, Fields Institute Monographs, vol. 35 (2017), Springer · Zbl 1387.60005 |
| [30] | Nica, A.; Speicher, R., Lectures on the Combinatorics of Free Probability (2006), Cambridge University Press · Zbl 1133.60003 |
| [31] | Ransford, T., Potential Theory in the Complex Plane (1995), Cambridge University Press · Zbl 0828.31001 |
| [32] | Rordam, M.; Larsen, F.; Laustsen, N., An Introduction to K-Theory for C*-Algebras (2000), Cambridge University Press · Zbl 0967.19001 |
| [33] | Rowen, L. H., Polynomial Identities in Ring Theory (1980), Academic Press · Zbl 0461.16001 |
| [34] | Schützenberger, M. P., On the definition of a family of automata, Inf. Control, 4, 245-270 (1961) · Zbl 0104.00702 |
| [35] | Schützenberger, M. P., On finite monoids having only trivial subgroups, Inf. Control, 8, 190-194 (1965) · Zbl 0131.02001 |
| [36] | Slinglend, N., NC Ball Maps and Change of Variables (2010), UCSD Phd Thesis in Mathematics |
| [37] | Speicher, R., Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc., 627 (1998) · Zbl 0935.46056 |
| [38] | Voiculescu, D. V.; Dykema, K.; Nica, A., Free Random Variables, CRM Monograph Series, vol. 1 (1992), AMS · Zbl 0795.46049 |
| [39] | Volcic, J., Matrix coefficient realization theory of noncommutative rational functions (2015), preprint |
| [40] | Volcic, J., On domains of noncommutative rational functions (2016), preprint |
| [41] | Yin, S., Non-commutative rational functions in strongly convergent random variables, Adv. Oper. Theory, 3, 1, 190-204 (2018) |
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