First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models

Communications in Mathematical Physics

Volumn 244, Issue 3, 2004, Pages 527-569

  Guionnet, Alice ^a  

^a UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1142276422     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0992-4     Document Type: Article

References (34)

1. Ben Arous, G., Guionnet, A.: Large deviations for Wigner's law and Voiculescu's non-commutative entropy. Prob. Th. Rel. Fields 108, 517-542 (1997)
2. Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42, 733-773 (1993)
3. Biane, P.: On the Free convolution with a Semi-circular distribution. Indiana Univ. Math. J. 46, 705-718 (1997)
4. Biane, P., Capitaine, M., Guionnet, A.: Large deviation bounds for matrix Brownian motion. Invent. Math. 152, 433-459 (2003)
5. Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure. Appl. Math. 52, 411-452 (1999)
6. Brézis, H.: Functional analysis. Paris: Masson, 1983
7. Cabanal-Duvillard, T., Guionnet, A.: Large deviations upper bounds and non commutative entropies for some matrices ensembles. Ann. Probab. 29, 1205-1261 (2001)
8. Cabanal-Duvillard, T., Guionnet, A.: Discussions around non-commutative entropies. Adv. Math. 174, 167-226 (2003)
9. Chadha, S., Madhoux, G., Mehta, M.L.: A method of integration over matrix variables II. J. Phys. A. 14, 579-586 (1981)
10. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95, 388-475 (1998)
11. Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Second edition, Berlin-Heidelberg-New York: Springer, 1998
12. Ercolani, N.M., McLaughlin, K.D.T-R.: Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration. To appear in Int. Math. Res. Notes, 2003
13. Eynard, B.: Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices. Nucl. Phys. B. 506, 633-664 (1997)
14. Eynard, B.: Random matrices, http://www-spht.cea.fr/lectures-notes.shtml
15. Guionnet, A.: Large deviation upper bounds and central limit theorems for band matrices. Ann. Inst. H. Poincaré Probab. Statist. 38, 341-384 (2002)
16. Guionnet, A., Zeitouni, O.: Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188, 461-515 (2002)
17. Guionnet, A. Zeitouni, O.: Addendum to: Large deviations asymptotics for spherical integrals. To appear in J. Funct. Anal. (2004)
18. Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457-485 (1986)
19. Loeper, G.: The inverse problem for the Euler-poisson system in cosmology. Preprint, 2003
20. Mahoux, G., Mehta, M.: A method of integration over matrix variables III. Indian J. Pure Appl. Math. 22, 531-546 (1991)
21. Matytsin, A.: On the large N-limit of the Itzykson-Zuber integral. Nucl. Phys. B411, 805-820 (1994)
22. Matytsin, A., Zaugg, P.: Kosterlitz-Thouless phase transitions on discretized random surfaces. Nucl. Phys. B497, 699-724 (1997)
23. Mehta, M.L.: Random matrices. 2nd ed., New York-London: Academic Press, 1991
24. Mehta, M.L.: A method of integration over matrix variables. Comm. Math. Phys. 79, 327-340 (1981)
25. Serre, D.: Sur le principe variationnel des équations de la mécanique des fluides parfaits. Math. Model. Num. Anal. 27, 739-758 (1993)
26. Szarek, S., Voiculescu, D.: Volumes of restricted Minkowsky Sums and the Free analogue of the Entropy Power Inequality. Commun. Math. Phys. 178, 563-570 (1996)
27. Tricomi, F.G.: Integral equations. New York: Interscience, 1957
28. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104, 201-220 (1991)
29. Voiculescu, D.: The analogues of Entropy and Fisher's Information Measure in Free Probability Theory, V: Noncommutative Hilbert Transforms. Invent. Math. 132, 189-227 (1998)
30. Voiculescu, D.: Lectures on free probability theory. In: Sptinger Lecture Notes Mathematics 1738, Berlin-Heidelberg-New York: Springer-Verlag, 2000, pp. 283-349
31. Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325-327 (1958)
32. Zinn-Justin, P.: Universality of correlation functions of hermitian random matrices in an external field. Commun. Math. Phys. 194, 631-650 (1998)
33. Zinn-Justin, P.: The dilute Potts model on random surfaces. J. Stat. Phys. 98, 245-264 (2000)
34. Zvonkin, A.: Matrix integrals and Map enumeration: an accessible introduction. Math. Comput. Mod. 26, 281-304 (1997)