8 research outputs found
Asymptotic expansion of a partition function related to the sinh-model
This paper develops a method to carry out the large- asymptotic analysis
of a class of -dimensional integrals arising in the context of the so-called
quantum separation of variables method. We push further ideas developed in the
context of random matrices of size , but in the present problem, two scales
and naturally occur. In our case, the equilibrium measure
is -dependent and characterised by means of the solution to a
Riemann--Hilbert problem, whose large- behavior is analysed in
detail. Combining these results with techniques of concentration of measures
and an asymptotic analysis of the Schwinger-Dyson equations at the
distributional level, we obtain the large- behavior of the free energy
explicitly up to . The use of distributional Schwinger-Dyson is a novelty
that allows us treating sufficiently differentiable interactions and the mixing
of scales and , thus waiving the analyticity assumptions
often used in random matrix theory.Comment: 158 pages, 4 figures (V2 introduction extended, missprints corrected,
clarifications added to lemma 3.1.9 and corollary 3.1.10
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
Asymptotic expansion of beta matrix models in the one-cut regime
We prove the existence of a 1/N expansion to all orders in beta matrix models
with a confining, off-critical potential corresponding to an equilibrium
measure with a connected support. Thus, the coefficients of the expansion can
be obtained recursively by the "topological recursion" of Chekhov and Eynard.
Our method relies on the combination of a priori bounds on the correlators and
the study of Schwinger-Dyson equations, thanks to the uses of classical complex
analysis techniques. These a priori bounds can be derived following Boutet de
Monvel, Pastur and Shcherbina, or for strictly convex potentials by using
concentration of measure. Doing so, we extend the strategy of Guionnet and
Maurel-Segala, from the hermitian models (beta = 2) and perturbative
potentials, to general beta models. The existence of the first correction in
1/N has been considered previously by Johansson and more recently by
Kriecherbauer and Shcherbina. Here, by taking similar hypotheses, we extend the
result to all orders in 1/N.Comment: 42 pages, 2 figures. v2: typos and a confusion of notation corrected.
v3: version to appear in Commun. Math. Phy
Spectral density asymptotics for Gaussian and Laguerre -ensembles in the exponentially small region
The first two terms in the large asymptotic expansion of the
moment of the characteristic polynomial for the Gaussian and Laguerre
-ensembles are calculated. This is used to compute the asymptotic
expansion of the spectral density in these ensembles, in the exponentially
small region outside the leading support, up to terms . The leading form
of the right tail of the distribution of the largest eigenvalue is given by the
density in this regime. It is demonstrated that there is a scaling from this,
to the right tail asymptotics for the distribution of the largest eigenvalue at
the soft edge.Comment: 19 page