69 research outputs found
Beyond universality in random matrix theory
In order to have a better understanding of finite random matrices with
non-Gaussian entries, we study the expansion of local eigenvalue
statistics in both the bulk and at the hard edge of the spectrum of random
matrices. This gives valuable information about the smallest singular value not
seen in universality laws. In particular, we show the dependence on the fourth
moment (or the kurtosis) of the entries. This work makes use of the so-called
complex Gaussian divisible ensembles for both Wigner and sample covariance
matrices.Comment: Published at http://dx.doi.org/10.1214/15-AAP1129 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Rigid C^*-tensor categories of bimodules over interpolated free group factors
Given a countably generated rigid C^*-tensor category C, we construct a
planar algebra P whose category of projections Pro is equivalent to C. From P,
we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid
C^*-tensor category Bim whose objects are bifinite bimodules over an
interpolated free group factor, and we show Bim is equivalent to Pro. We use
these constructions to show C is equivalent to a category of bifinite bimodules
over L(F_infty).Comment: 50 pages, many figure
Les effets du développement sur les politiques d’adoption des enfants : les cas de la Corée du Sud et du Vietnam
Le séisme dévastateur et meurtrier subi par Haïti en janvier 2010 a porté une nouvelle fois et brutalement sur le devant de la scène médiatique mondialisée la question de l’adoption d’enfants victimes du sous-développement : Est-ce une bonne réponse aux malheurs d’un pays pauvre ? Peut-elle régler les problèmes posés par l’enfance dans les pays du Tiers-monde ? Ne doit-on pas encadrer davantage l’adoption Internationale [André-Trevennec, 2008] Et chacun de prendre position pour ou contre l’adoption internationale, d’ériger en règle générale ou en loi commune tel ou tel cas de son entourage. Le regard de l’historien, s’appuyant sur des sources identifiées et une démarche construite permet de prendre du recul, de mettre en perspective les événements présents et passés [Denéchère, 2011]
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Open string theory and planar algebras
In this note we show that abstract planar algebras are algebras over the
topological operad of moduli spaces of stable maps with Lagrangian boundary
conditions, which in the case of the projective line are described in terms of
real rational functions. These moduli spaces appear naturally in the
formulation of open string theory on the projective line. We also show two
geometric ways to obtain planar algebras from real algebraic geometry, one
based on string topology and one on Gromov-Witten theory. In particular,
through the well known relation between planar algebras and subfactors, these
results establish a connection between open string theory, real algebraic
geometry, and subfactors of von Neumann algebras.Comment: 13 pages, LaTeX, 7 eps figure
Order in glassy systems
A directly measurable correlation length may be defined for systems having a
two-step relaxation, based on the geometric properties of density profile that
remains after averaging out the fast motion. We argue that the length diverges
if and when the slow timescale diverges, whatever the microscopic mechanism at
the origin of the slowing down. Measuring the length amounts to determining
explicitly the complexity from the observed particle configurations. One may
compute in the same way the Renyi complexities K_q, their relative behavior for
different q characterizes the mechanism underlying the transition. In
particular, the 'Random First Order' scenario predicts that in the glass phase
K_q=0 for q>x, and K_q>0 for q<x, with x the Parisi parameter. The hypothesis
of a nonequilibrium effective temperature may also be directly tested directly
from configurations.Comment: Typos corrected, clarifications adde
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
On Connected Diagrams and Cumulants of Erdos-Renyi Matrix Models
Regarding the adjacency matrices of n-vertex graphs and related graph
Laplacian, we introduce two families of discrete matrix models constructed both
with the help of the Erdos-Renyi ensemble of random graphs. Corresponding
matrix sums represent the characteristic functions of the average number of
walks and closed walks over the random graph. These sums can be considered as
discrete analogs of the matrix integrals of random matrix theory.
We study the diagram structure of the cumulant expansions of logarithms of
these matrix sums and analyze the limiting expressions in the cases of constant
and vanishing edge probabilities as n tends to infinity.Comment: 34 pages, 8 figure
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
Genus expansion for real Wishart matrices
We present an exact formula for moments and cumulants of several real
compound Wishart matrices in terms of an Euler characteristic expansion,
similar to the genus expansion for complex random matrices. We consider their
asymptotic values in the large matrix limit: as in a genus expansion, the terms
which survive in the large matrix limit are those with the greatest Euler
characteristic, that is, either spheres or collections of spheres. This
topological construction motivates an algebraic expression for the moments and
cumulants in terms of the symmetric group. We examine the combinatorial
properties distinguishing the leading order terms. By considering higher
cumulants, we give a central limit-type theorem for the asymptotic distribution
around the expected value
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