# SÃ©minaires

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

**Le : **22/02/2013 11h30

**Par : **Gaetan BOROT (UniversitÃ© de GenÃ¨ve)

**Lieu : **I 103

**Lien web : **

**RÃ©sumÃ© : **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.