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

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

The single ring theorem

We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where $U_n,V_n$ are independent Haar distributed on the orthogonal group.Comment: Correction of inadequate treatment of neighborhood of z=0 in original submission, various typos corrected following referee's remark

Large deviations of the extreme eigenvalues of random deformations of matrices

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)} X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.Comment: 44 page

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