[1] J. Wishart, “The generalized product moment distribution in samples from a normal multivariate population,” Biometrika, vol. 20A, no. 1/2, pp. 32–52, 1928.
[2] E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” The Annals of Mathematics, vol. 62, no. 3, pp. 548–564, 1955.
[3] D. Voiculescu, “Limit laws for random matrices and free products,” Inventiones mathe- maticae, vol. 104, no. 1, pp. 201–220, 1991.
[4] G. Akemann, J. Baik, and P. D. Francesco, The Oxford Handbook of Random Matrix Theory. Oxford Handbooks in Mathematics Series, Oxford University Press, 2015.
[5] B. Collins and I. Nechita, “Random matrix techniques in quantum information theory,” Journal of Mathematical Physics, vol. 57, no. 1, 2016.
[6] U. Haagerup and S. Thorbjørnsen, “A new application of random matrices: ext(c∗red(V2)) is not a group,” Annals of Mathematics, vol. 162, no. 2, pp. 711–775, 2005.
[7] B. Collins, A. Guionnet, and F. Parraud, “On the operator norm of non-commutative polynomials in deterministic matrices and iid gue matrices,” arXiv:1912.04588, 2019.
[8] F. Parraud, “On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices,” arXiv:2005.13834, 2020.
[9] B. Collins and C. Male, “The strong asymptotic freeness of Haar and deterministic ma- trices,” Annales Scientifiques de l’Ecole Normale Superieure, vol. 47, no. 1, pp. 147–163, 2014.
[10] F. Parraud, “Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid gue matrices,” arXiv:2011.04146, 2020.
[11] B. Collins and F. Parraud, “Concentration estimates for random subspaces of a tensor product, and application to quantum information theory,” arXiv:2012.00159, 2020.
[12] B. Collins, T. Mai, A. Miyagawa, F. Parraud, and S. Yin, “Convergence for non- commutative rational functions evaluated in random matrices,” arXiv:2103.05962, 2021.
[13] G. Anderson, A. Guionnet, and O. Zeitouni, An introduction to random matrices, vol. 118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2010.
[14] D. Voiculescu, “A strengthened asymptotic freeness result for random matrices with ap- plications to free entropy,” International Mathematics Research Notices, vol. 1998, no. 1, pp. 41–63, 1998.
[15] Z. Füredi and J. Komlós, “The eigenvalues of random symmetric matrices,” Combinator- ica, vol. 1, no. 3, pp. 233–241, 1981.
[16] Z.-D. Bai and Y.-Q. Yin, “Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix,” The Annals of Probability, vol. 16, no. 4, pp. 1729–1741, 1988.
[17] C. A. Tracy and H. Widom, “Level-spacing distributions and the Airy kernel,” Commu- nications in Mathematical Physics, vol. 159, no. 1, pp. 151–174, 1994.
[18] A. Soshnikov, “Universality at the edge of the spectrum in Wigner random matrices,” Communications in mathematical physics, vol. 207, no. 3, pp. 697–733, 1999.
[19] L. Erdős and H.-T. Yau, A dynamical approach to random matrix theory, vol. 28 of Courant Lecture Notes. American Mathematical Soc., 2017.
[20] T. Tao and V. Vu, “Random matrices: Universality of local eigenvalue statistics up to the edge,” Communications in Mathematical Physics, vol. 298, no. 2, pp. 549–572, 2010.
[21] J. O. Lee and J. Yin, “A necessary and sufficient condition for edge universality of Wigner matrices,” Duke Mathematical Journal, vol. 163, no. 1, pp. 117–173, 2014.
[22] L. Erdős, B. Schlein, and H.-T. Yau, “Wegner estimate and level repulsion for Wigner random matrices,” International Mathematics Research Notices, vol. 2010, no. 3, pp. 436– 479, 2010.
[23] A. Figalli and A. Guionnet, “Universality in several-matrix models via approximate trans- port maps,” Acta mathematica, vol. 217, no. 1, pp. 81–176, 2016.
[24] L. Erdős, T. Krüger, and Y. Nemish, “Local laws for polynomials of Wigner matrices,” Journal of Functional Analysis, vol. 278, no. 12, p. 108507, 2020.
[25] H. Schultz, “Non-commutative polynomials of independent Gaussian random matrices. the real and symplectic cases.,” Probability theory and related fields, vol. 131, no. 2, pp. 261–309, 2005.
[26] M. Capitaine and D. Martin, “Strong asymptotic freeness for Wigner and Wishart ma- trices,” Indiana university mathematics journal, vol. 56, no. 2, pp. 767–804, 2007.
[27] G. W. Anderson, “Convergence of the largest singular value of a polynomial in inde- pendent Wigner matrices,” The Annals of Probability, vol. 41, no. 3B, pp. 2103–2181, 2013.
[28] C. Male, “The norm of polynomials in large random and deterministic matrices,” Proba- bility Theory and Related Fields, vol. 154, no. 3-4, pp. 477–532, 2012.
[29] S. T. Belinschi and M. Capitaine, “Spectral properties of polynomials in independent Wigner and deterministic matrices,” Journal of Functional Analysis, vol. 273, no. 12, pp. 3901–3963, 2017.
[30] C. Stein, “A bound for the error in the normal approximation to the distribution of a sum of dependent random variables,” Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 583–602, 1972.
[31] G. Pisier, “Random matrices and subexponential operator spaces,” Israel Journal of Mathematics, vol. 203, no. 1, pp. 223–273, 2012.
[32] B. Hayes, “A random matrix approach to the Peterson-Thom conjecture,” arXiv:2008.12287, 2020.
[33] J. Harer and D. Zagier, “The euler characteristic of the moduli space of curves,” Inven- tiones mathematicae, vol. 85, no. 3, pp. 457–485, 1986.
[34] A. Zvonkin, “Matrix integrals and map enumeration: an accessible introduction,” Math- ematical and Computer Modelling, vol. 26, no. 8-10, pp. 281–304, 1997.
[35] G. ’t Hooft, “Magnetic monopoles in unified gauge theories,” Nuclear Physics: B, vol. 79, no. 2, pp. 276–284, 1974.
[36] E. Brézin, C. Itzykson, G. Parisi, and J. B. Zuber, “Planar diagrams,” Communications in Mathematical Physics, vol. 59, no. 1, pp. 35–51, 1978.
[37] N. M. Ercolani and K.-R. McLaughlin, “Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration,” International Mathematics Research Notices, vol. 2003, no. 14, pp. 755–820, 2003.
[38] S. Albeverio, L. Pastur, and M. Shcherbina, “On the 1/n expansion for some unitary invariant ensembles of random matrices,” Communications in Mathematical Physics, vol. 224, no. 1, pp. 271–305, 2001.
[39] A. Guionnet and E. Maurel-Segala, “Combinatorial aspects of matrix models,” Alea, vol. 1, pp. 241–279, 2006.
[40] A. Guionnet and E. Maurel-Segala, “Second order asymptotics for matrix models,” The Annals of Probability, vol. 35, no. 6, pp. 2160–2212, 2007.
[41] E. Maurel-Segala, “High order asymptotics of matrix models and enumeration of maps,” arXiv:math/0608192v1, 2006.
[42] B. Collins, A. Guionnet, and E. Maurel-Segala, “Asymptotics of unitary and orthogonal matrix integrals,” Advances in Mathematics, vol. 222, no. 1, pp. 172–215, 2009.
[43] A. Guionnet and J. Novak, “Asymptotics of unitary multimatrix models: The Schwinger– Dyson lattice and topological recursion,” Journal of Functional Analysis, vol. 268, no. 10, pp. 2851–2905, 2015.
[44] M. Shcherbina, “Asymptotic expansions for β-matrix models and their applications to the universality conjecture, random matrix theory, interacting particle systems, and inte- grable systems,” Mathematical Sciences Research Institute Publications, vol. 65, pp. 463– 482, 2014.
[45] L. O. Chekhov, B. Eynard, and O. Marchal, “Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach,” Theoretical and Mathematical Physics, vol. 166, no. 2, pp. 141–185, 2011.
[46] G. Borot and A. Guionnet, “Asymptotic expansion of β matrix models in the one-cut regime,” Communications in Mathematical Physics, vol. 317, no. 2, pp. 447–483, 2013.
[47] G. Borot and A. Guionnet, “Asymptotic expansion of β matrix models in the multi-cut regime,” arXiv:1303.1045, 2013.
[48] G. Borot, A. Guionnet, and K. K. Kozlowski, “Large-N asymptotic expansion for mean field models with Coulomb gas interaction,” International Mathematics Research Notices, vol. 2015, no. 20, pp. 10451–10524, 2015.
[49] G. Borot, A. Guionnet, and K. K. Kozlowski, Asymptotic expansion of a partition function related to the sinh-model. Mathematical Physics Studies, Springer, 2016.
[50] F. David, “Loop equations and non perturbative effects in two-dimensional quantum gravity,” Modern Physics Letters A, vol. 5, no. 13, pp. 1019–1029, 1990.
[51] V. A. Kazakov, “The appearance of matter fields from quantum fluctuations of 2d- gravity,” Modern Physics Letters A, vol. 4, no. 22, pp. 2125–2139, 1989.
[52] B. Eynard and N. Orantin, “Invariants of algebraic curves and topological expansion,” Communications in Number Theory and Physics, vol. 1, no. 2, pp. 347–452, 2007.
[53] B. Eynard, “Topological expansion for the 1-hermitian matrix model correlation func- tions,” Journal of High Energy Physics, vol. 2004, no. 11, p. 031, 2005.
[54] L. Chekhov and B. Eynard, “Matrix eigenvalue model: Feynman graph technique for all genera,” Journal of High Energy Physics, vol. 2006, no. 12, p. 026, 2006.
[55] G. Borot, B. Eynard, and N. Orantin, “Abstract loop equations, topological recursion and new applications,” Communications in Number Theory and Physics, vol. 9, no. 1, pp. 51–187, 2015.
[56] L. Chekhov and B. Eynard, “Hermitian matrix model free energy: Feynman graph tech- nique for all genera,” Journal of High Energy Physics, vol. 2006, no. 3, p. 014, 2006.
[57] B. Eynard and N. Orantin, “Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula,” Journal of High Energy Physics, vol. 2005, no. 12, p. 034, 2005.
[58] J. Ambjørn, L. Chekhov, C. Kristjansen, and Y. Makeenko, “Matrix model calculations beyond the spherical limit,” Nuclear Physics, Section B, vol. 404, no. 1-2, pp. 127–172, 1993.
[59] U. Haagerup and S. Thorbjørnsen, “Asymptotic expansions for the Gaussian unitary en- semble,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 15, no. 01, p. 1250003, 2012.
[60] A. Nica and R. Speicher, Lectures on the combinatorics of free probability, vol. 335 of London Mathematical Society Lecture Note Series. Cambridge University Press, 2006.
[61] M. B. Hastings, “Superadditivity of communication capacity using entangled inputs,” Nature Physics, vol. 5, no. 4, pp. 255–257, 2009.
[62] P. Hayden and A. Winter, “Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1,” Communications in mathematical physics, vol. 284, no. 1, pp. 263–280, 2008.
[63] F. G. Brandao and M. Horodecki, “On Hastings’ counterexamples to the minimum output entropy additivity conjecture,” Open Systems & Information Dynamics, vol. 17, no. 01, pp. 31–52, 2010.
[64] S. T. Belinschi, B. Collins, and I. Nechita, “Almost one bit violation for the additivity of the minimum output entropy,” Communications in Mathematical Physics, vol. 341, no. 3, pp. 885–909, 2016.
[65] G. Aubrun, S. Szarek, and E. Werner, “Hastings’s additivity counterexample via Dvoret- zky’s theorem,” Communications in mathematical physics, vol. 305, no. 1, pp. 85–97, 2011.
[66] B. Collins, “Haagerup’s inequality and additivity violation of the minimum output en- tropy,” Houston Journal of Mathematics, vol. 44, no. 1, pp. 253–261, 2016.
[67] S. Belinschi, B. Collins, and I. Nechita, “Eigenvectors and eigenvalues in a random sub- space of a tensor product,” Inventiones mathematicae, vol. 190, no. 3, pp. 647–697, 2012.
[68] T. Mai, R. Speicher, and S. Yin, “The free field: zero divisors, Atiyah property and realizations via unbounded operators,” arXiv:1805.04150, 2018.
[69] T. Mai, R. Speicher, and S. Yin, “The free field: realization via unbounded operators and Atiyah property,” arXiv:1905.08187, 2019.
[70] T. Cabanal-Duvillard, “Fluctuations de la loi empirique de grandes matrices aléatoires,” in Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 37, pp. 373–402, 2001.
[71] A. Guionnet, “Large deviations upper bounds and central limit theorems for non- commutative functionals of Gaussian large random matrices,” in Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 38, pp. 341–384, 2002.
[72] J. A. Mingo and R. Speicher, “Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces,” Journal of Functional Analysis, vol. 235, no. 1, pp. 226–270, 2006.
[73] P. Biane and R. Speicher, “Free diffusions, free entropy and free Fisher information,” in Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 37, pp. 581–606, 2001.
[74] D. Shlyakhtenko and P. Skoufranis, “Freely independent random variables with non- atomic distributions,” Transactions of the American Mathematical Society, vol. 367, no. 9, pp. 6267–6291, 2015.
[75] N. P. Brown and N. Ozawa, ∗-Algebras and Finite-Dimensional Approximations, vol. 88 of Graduate studies in mathematics.
[76] G. J. Murphy, C∗-algebras and operator theory. Elsevier Science, 1990.
[77] A. Guionnet, “Large random matrices: Lectures on macroscopic asymptotics,” Ecole d’Eté de Probabilités de Saint-Flour XXXVI–2006, 2009.
[78] G. Pisier, Introduction to operator space theory. No. 294, Cambridge University Press, 2003.
[79] J. A. Mingo, P. Śniady, and R. Speicher, “Second order freeness and fluctuations of random matrices: II. unitary random matrices,” Advances in Mathematics, vol. 209, no. 1, pp. 212–240, 2007.
[80] A. Guionnet, Asymptotics of Random Matrices and Related Models: The Uses of Dyson- Schwinger Equations, vol. 130. American Mathematical Society, 2019.
[81] P. Biane and R. Speicher, “Stochastic calculus with respect to free Brownian motion and analysis on Wigner space,” Probability theory and related fields, vol. 112, no. 3, pp. 373– 409, 1998.
[82] Y. Dabrowski, “A free stochastic partial differential equation,” in Annales de l’IHP Prob- abilités et statistiques, vol. 50, pp. 1404–1455, 2014.
[83] P. Biane, “Segal–Bargmann transform, functional calculus on matrix spaces and the the- ory of semi-circular and circular systems,” Journal of Functional Analysis, vol. 144, no. 1, pp. 232–286, 1997.
[84] B. Collins, A. Dahlqvist, and T. Kemp, “Strong convergence of unitary Brownian motion,” Probability Theory and Related Fields, vol. 170, no. 1-2, pp. 49–93, 2015.
[85] T. Lévy and M. Maïda, “Central limit theorem for the heat kernel measure on the unitary group,” Journal of Functional Analysis, vol. 259, no. 12, pp. 3163–3204, 2010.
[86] G. Cébron and T. Kemp, “Fluctuations of Brownian motions on GLN ,” arXiv:1409.5624, 2014.
[87] E. A. Nikitopoulos, “Noncommutative Ck functions and Fréchet derivatives of operator functions,” arXiv:2011.03126, 2020.
[88] P. Biane, “Free Brownian motion, free stochastic calculus and random matrice, in free probability theory,” Fields Institute Communications, vol. 12, pp. 1–19, 1997.
[89] F. Bekerman, A. Figalli, and A. Guionnet, “Transport maps for β-matrix models and universality,” Communications in mathematical physics, vol. 338, no. 2, pp. 589–619, 2015.
[90] B. Collins and S.-G. Youn, “Additivity violation of the regularized minimum output entropy,” arXiv:1907.07856, 2019.
[91] R. T. Rockafellar, Convex analysis, vol. 36. Princeton university press, 1970.
[92] D. Voiculescu, K. J. Dykema, and A. Nica, Free random variables. No. 1, American Mathematical Society, 1992.
[93] B. Collins and I. Nechita, “Random quantum channels I: Graphical calculus and the bell state phenomenon,” Communications in Mathematical Physics, vol. 297, no. 2, pp. 345– 370, 2010.
[94] J. A. Mingo and R. Speicher, Free probability and random matrices, vol. 35 of Fields Institute Monographs. Springer, 2017.
[95] S. Yin, “Non-commutative rational functions in strong convergent random variables,” Advances in Operator Theory, vol. 3, no. 1, pp. 178–192, 2018.
[96] S. A. Amitsur, “Rational identities and applications to algebra and geometry,” Journal of Algebra, vol. 3, no. 3, pp. 304–359, 1966.
[97] P. M. Cohn, Free ideal rings and localization in general rings, vol. 3. Cambridge university press, 2006.
[98] P. A. Linnell, “Division rings and group von Neumann algebras,” in Forum Math, vol. 5, pp. 561–576, 1993.
[99] L. Erdős, T. Krüger, and Y. Nemish, “Scattering in quantum dots via noncommutative rational functions,” arXiv:1911.05112, 2019.
[100] Z. Ma and F. Yang, “Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations,” arXiv:2102.03297, 2021.
[101] R. Speicher, “Regularity of non-commutative distributions and random matrices,” in Publications of MFO, OWR-2019-56, pp. 3513–3514, 2019.
[102] S. Yin, “On the rational functions in non-commutative random variables,” 2020.
[103] G. Duchamp and C. Reutenauer, “Un critere de rationalité provenant de la géométrie non commutative,” Inventiones Mathematicae, vol. 128, no. 3, pp. 613–622, 1997.
[104] P. A. Linnell, “A rationality criterion for unbounded operators,” Journal of Functional Analysis, vol. 171, no. 1, pp. 115–123, 2000.
[105] A. Connes, Noncommutative geometry. Transl. from the French by Sterling Berberian. San Diego, CA: Academic Press, 1994.
[106] A. Miyagawa, “The estimation of non-commutative derivatives and the asymptotics for the free field in free probability theory,” Master’s thesis, Kyoto University, 2021.
[107] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, “Noncommutative rational functions, their difference-differential calculus and realizations,” Multidimensional Systems and Sig- nal Processing, vol. 23, no. 1-2, pp. 49–77, 2012.
[108] P. Hrubes and A. Wigderson, “Non-commutative arithmetic circuits with division,” in Proceedings of the 5th conference on Innovations in theoretical computer science, pp. 49– 66, 2014.
[109] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, “Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting,” Linear Alge- bra and its Applications, vol. 430, no. 4, pp. 869–889, 2009.
[110] J. W. Helton, T. Mai, and R. Speicher, “Applications of realizations (aka linearizations) to free probability,” Journal of Functional Analysis, vol. 274, no. 1, pp. 1–79, 2018.
[111] P. M. Cohn and C. Reutenauer, “A normal form in free fields,” Canadian Journal of Mathematics, vol. 46, no. 3, pp. 517–531, 1994.
[112] P. M. Cohn and C. Reutenauer, “On the construction of the free field,” International journal of Algebra and Computation, vol. 9, no. 3-4, pp. 307–323, 1999.
[113] C. Vargas, “A general solution to (free) deterministic equivalents,” Contributions of Mex- ican Mathematicians Abroad in Pure and Applied Mathematics, vol. 709, p. 131, 2018.
[114] F. J. Murray and J. von Neumann, “On rings of operators,” Annals of Mathematics, vol. 37, no. 1, pp. 116–229, 1936.
[115] B. Blackadar, Operator algebras: theory of ∗-algebras and von Neumann algebras, vol. 122. Berlin:Springer, 2006.
[116] M. Terp, “lp spaces associated with von Neumann algebras,” Math. Institute, Copenhagen Univ., vol. 3, no. 4, 1981.
[117] J. B. Conway, A course in functional analysis. New York etc.: Springer–Verlag, 1990.
[118] A. Connes and D. Shlyakhtenko, “l2-homology for von Neumann algebras,” Journal für die reine und angewandte Mathematik, vol. 2005, no. 586, pp. 125–168, 2005.
[119] D. Voiculescu, “The analogues of entropy and of Fisher’s information measure in free probability theory, II,” Inventiones mathematicae, vol. 118, no. 1, pp. 411–440, 1994.
[120] J. Volcic, “Matrix coefficient realization theory of noncommutative rational functions,” Journal of Algebra, vol. 499, pp. 397–437, 2018.
[121] J. Volcic, “Hilbert’s 17th problem in free skew fields,” arXiv:2101.02314, 2021.
[122] H. Bercovici and D. Voiculescu, “Free convolution of measures with unbounded support,” Indiana University Mathematics Journal, vol. 42, no. 3, pp. 733–773, 1993.