138 research outputs found
On Classical Analogues of Free Entropy Dimension
We define a classical probability analogue of Voiculescu's free entropy
dimension that we shall call the classical probability entropy dimension of a
probability measure on . We show that the classical probability
entropy dimension of a measure is related with diverse other notions of
dimension. First, it can be viewed as a kind of fractal dimension. Second, if
one extends Bochner's inequalities to a measure by requiring that microstates
around this measure asymptotically satisfy the classical Bochner's
inequalities, then we show that the classical probability entropy dimension
controls the rate of increase of optimal constants in Bochner's inequality for
a measure regularized by convolution with the Gaussian law as the
regularization is removed. We introduce a free analogue of the Bochner
inequality and study the related free entropy dimension quantity. We show that
it is greater or equal to the non-microstates free entropy dimension
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
Moderate deviations for the spectral measure of certain random matrices
We derive a moderate deviations principle for matrices of the form XN = DN + WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit µD. Our techniques are based on a dynamical approach introduced by Cabanal-Duvillard and Guionnet
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
Eigenvalue variance bounds for Wigner and covariance random matrices
This work is concerned with finite range bounds on the variance of individual
eigenvalues of Wigner random matrices, in the bulk and at the edge of the
spectrum, as well as for some intermediate eigenvalues. Relying on the GUE
example, which needs to be investigated first, the main bounds are extended to
families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment
Theorem and recent localization results by Erd\"os, Yau and Yin. The case of
real Wigner matrices is obtained from interlacing formulas. As an application,
bounds on the expected 2-Wasserstein distance between the empirical spectral
measure and the semicircle law are derived. Similar results are available for
random covariance matrices
Limiting dynamics for spherical models of spin glasses at high temperature
We analyze the coupled non-linear integro-differential equations whose
solutions is the thermodynamical limit of the empirical correlation and
response functions in the Langevin dynamics for spherical p-spin disordered
mean-field models. We provide a mathematically rigorous derivation of their FDT
solution (for the high temperature regime) and of certain key properties of
this solution, which are in agreement with earlier derivations based on
physical grounds
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
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