113 research outputs found
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 with a covariance kernel
and boundary condition . We study the long time behaviour of
as the time parameters go to infinity, according to the asymptotic
behaviour of . 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 -ensembles
We define a new diffusive matrix model converging towards the -Dyson
Brownian motion for all that provides an explicit construction
of -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 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
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