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Small dispersion regime in integrable systems, random matrices and topological recursion

Le : 22/02/2013 11h30
Par : Gaetan BOROT (Université de Genève)
Lieu : I 103
Lien web :
Résumé : Integrable systems in Lax form are hierarchies of non-linear PDEs (or difference equation) which can be written as compatibility equations of a system of ODE's (or difference equation). An example is given by the Toda chain, which is a equation of evolution for two sequences (u_n(t),v_n(t)) depending on a infinite number of times. For certain initial condition their solutions are closely related to recurrence coefficients of orthogonal polynomials on the real line, and thus to random hermitian matrices. The problem of studying the continuum limit of the Toda chain is mapped to the problem of studying the large size asymptotics in random matrices. It is known that, depending on the number (g + 1) of segments of the support of a certain probability measure constructed from initial data (defining a spectral curve of genus g), the solution will feature fast oscillations whose wavelength is of order of the step of the lattice, describe by genus g theta functions. Powerful Riemann-Hilbert techniques have been developed, allowing in particular the full description of the leading behavior of solutions of Toda chain. However, carrying the RH analysis beyond the leading order, although possible in principle, is very cumbersome. In a recent work with Guionnet which I will present, we developed purely probabilistic techniques to establish the all-order asymptotic expansion in the 1-hermitian matrix model in the multi-cut regime, justifying in this way heuristic arguments proposed formerly by Bonnet, David and Eynard. As a consequence, we can prove the all-order asymptotics of solutions of the Toda chain in the continuum limit in (g + 1)-cut regime for any g. The same asymptotic phenomenon are expected for solutions of a more general class of integrable systems (those of topological type, as studied by Dubrovin). In a second part of the talk, I will describe and motivate a conjecture we proposed with Eynard, using the topological recursion developed by Eynard and Orantin, which associates to any spectral curve of genus g, an asymptotic series for the Tau function of a (yet to determine) integrable system in Lax form. Here, the small parameter of expansion is a dispersion parameter h, which is added in front of each derivative of the integrable system. The rigorous results about multi-cut regime in matrix models described in the first part of the talk justify this conjecture in a special case.