Schwinger-Dyson equations : classical and quantum

• Autorzy: Mingo J. A. , Speicher R.
• Języki publikacji: EN

Abstrakty

EN

In this note we want to have another look on Schwinger-Dyson equations for the eigenvalue distributions and the fluctuations of classical unitarily invariant random matrix models. We are exclusively dealing with one-matrix models, for which the situation is quite well understood. Our point is not to add any new results to this, but to have a more algebraic point of view on these results and to understand from this perspective the universality results for fluctuations of these random matrices. We will also consider corresponding non-commutative or “quantum” random matrix models and contrast the results for fluctuations and Schwinger-Dyson equations in the quantum case with the findings from the classical case.

Słowa kluczowe

EN

free probability; random matrices; Schwinger-Dyson equation

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 275--285
• Opis fizyczny: Bibliogr. 14 poz., rys.

Twórcy

autor

Mingo J. A.

• Queen’s University, Department of Mathematics and Statistics, Kingston, Ontario, K7L 3N6, Canada

autor

Speicher R.

• Universität des Saarlandes, FR 6.1-Mathematik, Postfach 15 11 50, D-66123 Saarbrücken, Germany

Bibliografia

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• [2] S. Curran and R. Speicher, Quantum invariant families of matrices in free probability, J. Funct. Anal. 261 (2011), pp. 897-933.
• [3] B. Eynard, Random Matrices, Cours de Physique Théorique de Saclay, 2000.
• [4] A. Guionnet, Large Random Matrices: Lectures on Macroscopic Asymptotics (École d’Été de Probabilités de Saint-Flour XXXVI, 2006), Lecture Notes in Math., Vol 1957, Springer, 2009.
• [5] A. Guionnet and E. Maurel-Segala, Second order asymptotics for matrix models, Ann. Probab. 35 (2007), pp. 2160-2212.
• [6] K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), pp. 151-204.
• [7] T. Kusalik, J. Mingo, and R. Speicher, Orthogonal polynomials and fluctuations of random matrices, J. Reine Angew. Math. (Crelle’s J.) 604 (2007), pp. 1-46.
• [8] J. Mingo, P. Sniady, and R. Speicher, Second order freeness and fluctuations of random matrices; II. Unitary random matrices, Adv. Math. 209 (2007), pp. 212-240.
• [9] J. Mingo, R. Speicher, and E. Tan, Second order cumulants of products, Trans. Amer. Math. Soc. 361 (2009), pp. 4751-4781.
• [10] A. Nica, D. Shlyakhtenko, and R. Speicher, R-cyclic families of random matrices in free probability, J. Funct. Anal. 188 (2002), pp. 227-271.
• [11] A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, Cambridge University Press, 2006.
• [12] D. Politzer, Random-matrix description of the distribution of mesoscopic conductance, Phys. Rev. B 40 (1989), pp. 11917-11919.
• [13] D. Voiculescu, A note on cyclic gradient, Indiana Univ. Math. J. 49 (2000), pp. 837-841.
• [14] D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monogr. Ser., Amer. Math. Soc., 1992.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).