Asymptotic expansion of beta matrix models in the multi-cut regime

We push further our study of the all-order asymptotic expansion in beta matrix models with a confining, offcritical potential, in the regime where the support of the equilibrium measure is a reunion of segments. We first address the case where the filling fractions of those segments are fixed, and show the existence of a 1/N expansion to all orders. Then, we study the asymptotic of the sum over filling fractions, in order to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. We describe the application of our results to study the all-order small dispersion asymptotics of solutions of the Toda chain related to the one hermitian matrix model (beta = 2) as well as orthogonal polynomials outside the bulk.Comment: 59 pages. v4: proof of smooth dependence in filling fraction (Appendix A) corrected, comment on the analogue of the CLT added, typos corrected. v5: Section 7 completely rewritten, interpolation for expansion of partition function is now done by decoupling the cuts, details on comparison to Eynard-Chekhov coefficients added in the introductio

Long time behavior of the solutions to non-linear Kraichnan equations

We consider the solution of a nonlinear Kraichnan equation $$\partial_s H(s,t)=\int_t^s H(s,u)H(u,t) k(s,u) du,\quad s\ge t$$ with a covariance kernel $k$ and boundary condition $H(t,t)=1$. We study the long time behaviour of $H$ as the time parameters $t,s$ go to infinity, according to the asymptotic behaviour of $k$. This question appears in various subjects since it is related with the analysis of the asymptotic behaviour of the trace of non-commutative processes satisfying a linear differential equation, but also naturally shows up in the study of the so-called response function and aging properties of the dynamics of some disordered spin systems.Comment: 32 page

A diffusive matrix model for invariant β\beta-ensembles

We define a new diffusive matrix model converging towards the $\beta$-Dyson Brownian motion for all $\beta\in [0,2]$ that provides an explicit construction of $\beta$-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when $\beta< 1$ and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues

Second order asymptotics for matrix models

We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one. In order to do that, we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges.Comment: Published in at http://dx.doi.org/10.1214/009117907000000141 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org