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Voiculescu’s Entropy and Potential Theory
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 57-69.

Nous donnons une démonstration nouvelle, s’appuyant sur des inégalités polynomiales et certains aspects de la théorie du potentiel, des résultats de grande déviation pour des ensembles de matrices hermitiennes aléatoires.

We give a new proof, relying on polynomial inequalities and some aspects of potential theory, of large deviation results for ensembles of random hermitian matrices.

DOI : 10.5802/afst.1305
Thomas Bloom 1

1 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3 Canada
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     title = {Voiculescu{\textquoteright}s {Entropy} and {Potential} {Theory}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {57--69},
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Thomas Bloom. Voiculescu’s Entropy and Potential Theory. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 57-69. doi : 10.5802/afst.1305. https://afst.centre-mersenne.org/articles/10.5802/afst.1305/

[Ba] Baran (M.).— Complex equilibrium measure and Bernstein type theorems for compact sets in n , Proc Am. Math. Soc. 123. no. 2, p. 485-494 (1995). | MR | Zbl

[Be1] Berman (R.).— Large deviations and entropy for determinental point processes on complex manifolds, arxiv:0812.4224.

[Be2] Berman (R.).— Determinental point processes and fermions on complex manifolds: bulk universality arxiv:0811.3341.

[B1-Le1] Bloom (T.) and Levenberg (N.).— Capacity convergence results and applications to a Bernstein-Markov inequality, Tr. Am. Math Soc. 351. no. 12, p. 4753-4767 (1999). | MR | Zbl

[B1-Le2] Bloom (T.) and Levenberg (N.).— Asymptotics for Christoffel functions of Planar Measures, J. D’Anal Math. 106, p. 353-371 (2008). | MR | Zbl

[B1-Le3] Bloom (T.) and Levenberg (N.).— Transfinite diameter notions in n and integrals of Van-DerMonde determinants, arxiv:0712.2844. | Zbl

[B1] Bloom (T.).— Weighted polynomials and weighted pluripotential theory, Tr. Am. Math Soc. 361 no. 4, p. 2163-2179 (2009). | MR | Zbl

[B1, talk] Bloom (T.).— “Large Deviations for VanDerMonde determinants" talk given at the Work-shop on Complex Hyperbolic Geometry and Related Topics (November 17-21, 2008) at the Fields Institute. http://www.fields.utoronto.ca/audio/08-09/hyperbolic/bloom/.

[Be-Gu] Ben Arous (G.) and Guionnet (A.).— Large deviation for Wigner’s law and Voiculescu’s non-commutative entopy, Prob. Th. Related Fields 108 (1997), p. 517-542. | MR | Zbl

[Be-Ze] Ben Arous (G.) and Zeitouni (O.).— Large deviations from the circular law, ESAIM: Probability and Statistics 2, 123-134 (1998). | Numdam | MR | Zbl

[Bo-Er] Borwein (P.) and Erdelyi (T.).— Polynomials and Polynomial Inequalities, Springer Graduate Texts in Mathematics 161, New York (1995). | MR | Zbl

[De-Ze] Dembo (A.) and Zeitouni (O.).— Large Deviation Techniques and Applications, 2nd edition, Springer, New York (1998). | MR | Zbl

[Dei] Deift (P.).— Othogonal Polynomials and Random Matrices: A Riemann-Hilbert approach, AMS Providence RI (1999). | MR | Zbl

[El] Ellis (R.S.).— Entropy, Large Deviations and Statistical Mechanics, Springer, New York/Berlin (1985). | MR | Zbl

[Hi-Pe] Hiai (F.) and Petz (D.).— The Semicircle Law, Free Random Variables and Entropy, AMS Providence RI (2000). | MR | Zbl

[Kl] Klimek (M.).— Pluripotential Theory, Oxford University Press, Oxford (1991). | MR | Zbl

[Pl] Plesniak (W.).— Inegalite de Markov en plusieurs variables, Int. J of Math and Math. Science, Art ID 24549, p. 1-12 (2006). | MR | Zbl

[Sa-To] Saff (E.) and Totik (V.).— Logarithmic Potential with External Fields, Springer-Verlag, Berlin (1997). | MR | Zbl

[St-To] Stahl (H.) and Totik (V.).— General Orthogonal Polynomials, Cambridge Unviersity Press, Cambridge (1992). | MR | Zbl

[Vo1] Voiculescu (D.).— The analogues of entropy and of Fisher’s information measure in free probability theory I, Comm. Math. Phy. 155, p. 71-92 (1993). | MR | Zbl

[Vo2] Voiculescu (D.).— The analogues of entropy and of Fisher’s information measure in free probability II, Inv. Math. 118, p. 411-440 (1994). | MR | Zbl

[Ze-Ze] Zeitouni (O.) and Zelditch (S.).— Large Deviations of empirical zero point measures on Riemann surfaces, I : g = 0 arxiv:0904.4271.

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