Free Probability, Random Matrices, and Representations of Non-commutative Rational Functions

  • Tobias Mai
  • Roland SpeicherEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


A fundamental problem in free probability theory is to understand distributions of “non-commutative functions” in freely independent variables. Due to the asymptotic freeness phenomenon, which occurs for many types of independent random matrices, such distributions can describe the asymptotic eigenvalue distribution of corresponding random matrix models when their dimension tends to infinity. For non-commutative polynomials and rational functions, an algorithmic solution to this problem is presented. It relies on suitable representations for these functions.



This work was supported by the ERC Advanced Grant “Non-commutative Distributions in Free Probability” (grant no. 339760).


  1. 1.
    Amitsur, S.A.: Rational identities and applications to algebra and geometry. J. Algebra 3, 304–359 (1966)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Anderson, G.W.: Convergence of the largest singular value of a polynomial in independent Wigner matrices. Ann. Probab. 41(3B), 2103–2181 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ball, J.A., Groenewald, G., Malakorn, T.: Structured noncommutative multidimensional linear systems. SIAM J. Control Optim. 44(1), 1474–1528 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Belinschi, S.T., 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. 732, 21–53 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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. Linear Algebra Appl. 537, 48–83 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bergman, G.W.: Skew fields of noncommutative rational functions (preliminary version). Séminaire Schützenberger 1, 1–18 (1969–1970)Google Scholar
  7. 7.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. Springer, Berlin (1988)CrossRefGoogle Scholar
  8. 8.
    Biane, P., Lehner, F.: Computation of some examples of Brown’s spectral measure in free probability. Colloq. Math. 90(2), 181–211 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brown, L.G.: Lidskii’s theorem in the type II case. In: Araki, H. (ed.) Geometric Methods in Operator Algebras, Proceedings of the US-Japan Seminar, Kyoto 1983. Pitman Research Notes in Mathematics Series, vol. 123, pp. 1–35. Longman Scientific and Technical/Wiley, New York/Harlow (1986)Google Scholar
  10. 10.
    Cohn, P.M.: Free Rings and Their Relations, 2nd edn. London Mathematical Society Monographs, vol. 19. Academic Press, London (Harcourt Brace Jovanovich, Publishers), XXII, p. 588 (1985)Google Scholar
  11. 11.
    Cohn, P.M.: Free Ideal Rings and Localization in General Rings. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  12. 12.
    Cohn, P.M., Reutenauer, C.: A normal form in free fields. Can. J. Math. 46(3), 517–531 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cohn, P.M., Reutenauer, C.: On the construction of the free field. Int. J. Algebra Comput. 9(3–4), 307–323 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fliess, M.: Sur divers produits de séries formelles. Bull. Soc. Math. Fr. 102, 181–191 (1974)CrossRefGoogle Scholar
  15. 15.
    Füredi, Z., Komloś, J.: The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Haagerup, U., Larsen, F.: Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176(2), 331–367 (2000)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Haagerup, U., Schultz, H., Thorbjørnsen, S.: A random matrix approach to the lack of projections in \(C_{\mathrm {red}}^{\ast }(\mathbb F_2)\). Adv. Math. 204(1), 1–83 (2006)Google Scholar
  18. 18.
    Haagerup, U., Thorbjørnsen, S.: A new application of random matrices: \(\operatorname {Ext} (C_{\text{red}}^*(F_2))\) is not a group. Ann. Math. (2) 162(2), 711–775 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Helton, J.W., Mai, T., Speicher, R.: Applications of realizations (aka Linearizations) to free probability. J. Funct. Anal. 274(1), 1–79 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Helton, J.W., McCullough, S.A., Vinnikov, V.: Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240(1), 105–191 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. American Mathematical Society (AMS), Providence (2000)Google Scholar
  22. 22.
    Higman, G.: The units of group-rings. Proc. Lond. Math. Soc. (2) 46, 231–248 (1940)CrossRefGoogle Scholar
  23. 23.
    Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting. Linear Algebra Appl. 430(4), 869–889 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Noncommutative rational functions, their difference-differential calculus and realizations. Multidim. Syst. Signal Process. 23(1–2), 49–77 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Noncommutative Function Theory. American Mathematical Society (AMS), Providence (2014)zbMATHGoogle Scholar
  26. 26.
    Kalman, R.E.: Mathematical description of linear dynamical systems. J. Soc. Ind. Appl. Math. Ser. A Control 1, 152–192 (1963)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kalman, R.E.: Realization theory of linear dynamical systems. In: Control Theory and Topics in Functional Analysis, vol. II. Lecture Presented at the International Seminar Course, Trieste, vol. 1974, pp. 235–256 (1976)Google Scholar
  28. 28.
    Kleene, S.C.: Representation of Events in Nerve Nets and Finite Automata. Automata Studies, p. 341. Princeton University Press, Princeton (1956)Google Scholar
  29. 29.
    Malcolmson, P.: A prime matrix ideal yields a skew field. J. Lond. Math. Soc. II Ser. 18, 221–233 (1978)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mingo, J.A., Speicher, R.: Free Probability and Random Matrices. Fields Institute Monographs, vol. 35. Springer, New York (2017)CrossRefGoogle Scholar
  31. 31.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  32. 32.
    Schützenberger, M.P.: On the definition of a family of automata. Inf. Control 4, 245–270 (1961)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Taylor, J.L.: A general framework for a multi-operator functional calculus. Adv. Math. 9, 183–252 (1972)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Taylor, J.L.: Functions of several noncommuting variables. Bull. Am. Math. Soc. 79, 1–34 (1973)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Voiculescu, D., Dykema, K.J., Nica, A.: Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. American Mathematical Society, Providence (1992)Google Scholar
  37. 37.
    Volcic, J.: Matrix coefficient realization theory of noncommutative rational functions. J. Algebra 499, 397–437 (2018)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. (2) 62, 548–564 (1955)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yin, S.: Non-commutative rational functions in strongly convergent random variables. Adv. Oper. Theory 3(1), 190–204 (2018)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Fachrichtung MathematikUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations