- The study of loop equations (also called Pastur equations or Schwinger-Dyson equations) is one (of the) powerful technique(s) to determine all order asymptotics of "matrix models", for large size N of the matrices. J.Ambjorn, L.Chekhov, C.Kristjansen, Yu.Makeenko, G.Akemann contributed to the development of this technique. For hermitean matrix models, the general solution of loop equations was found by B. Eynard in 2004-2005 in terms of algebraic geometry of a certain plane curve, related to the density of eigenvalues at large N. Though this construction does not use orthogonal polynomials or integrability, it is deeply related with those structures. Then, it was extended in 2006 by L.Chekhov and B.Eynard to the beta ensembles.
In this talk, I shall illustrate this method (called topological recursion) on the simple case of beta ensembles on the real line with a hard edge a, in the one-cut regime. Actually, the partition function is the probability that the maximum eigenvalue in beta ensembles is smaller that a. Its leading behavior at large N is the large deviation function for the maximum eigenvalue, and the subleading orders give corrections to the large deviations.
In a joint work with B.Eynard, S.Majumdar and C.Nadal (math-ph/1009.1945), we have obtained explicit expressions for the "dominant terms" of the large deviation function, and probed them numerically. I will also discuss the connection to the left tail of Tracy-Widom laws. This leads to recover known results for the cases of GOE/GUE/GSE, to obtain some new results for general beta, and to discuss universality.