178 research outputs found
An introduction to random matrix theory
These are lectures notes for a 4h30 mini-course held in Ulaanbaatar, National
University of Mongolia, August 5-7th 2015, at the summer school "Stochastic
Processes and Applications". It aims at presenting an introduction to basic
results of random matrix theory and some of its motivations, targeted to a
large panel of students coming from statistics, finance, etc. Only a small
background in probability is required.Comment: 55 pages, 10 figure
Planckian Axions in String Theory
We argue that super-Planckian diameters of axion fundamental domains can
naturally arise in Calabi-Yau compactifications of string theory. In a theory
with axions , the fundamental domain is a polytope defined by the
periodicities of the axions, via constraints of the form . We compute the diameter of the fundamental domain in terms of
the eigenvalues of the metric on field space, and also,
crucially, the largest eigenvalue of . At large ,
approaches a Wishart matrix, due to universality, and we show that
the diameter is at least , exceeding the naive Pythagorean range by a
factor . This result is robust in the presence of constraints,
while for the diameter is further enhanced by eigenvector delocalization
to . We directly verify our results in explicit Calabi-Yau
compactifications of type IIB string theory. In the classic example with
where parametrically controlled moduli stabilization was
demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys . The random matrix analysis then predicts, and we
exhibit, axion diameters for the precise vacuum parameters found in
[1]. Our results provide a framework for achieving large-field axion inflation
in well-understood flux vacua.Comment: 42 pages, 4 figure
Random matrix techniques in quantum information theory
The purpose of this review article is to present some of the latest
developments using random techniques, and in particular, random matrix
techniques in quantum information theory. Our review is a blend of a rather
exhaustive review, combined with more detailed examples -- coming from research
projects in which the authors were involved. We focus on two main topics,
random quantum states and random quantum channels. We present results related
to entropic quantities, entanglement of typical states, entanglement
thresholds, the output set of quantum channels, and violations of the minimum
output entropy of random channels
Orthogonal polynomial ensembles in probability theory
We survey a number of models from physics, statistical mechanics, probability
theory and combinatorics, which are each described in terms of an orthogonal
polynomial ensemble. The most prominent example is apparently the Hermite
ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE),
and other well-known ensembles known in random matrix theory like the Laguerre
ensemble for the spectrum of Wishart matrices. In recent years, a number of
further interesting models were found to lead to orthogonal polynomial
ensembles, among which the corner growth model, directed last passage
percolation, the PNG droplet, non-colliding random processes, the length of the
longest increasing subsequence of a random permutation, and others. Much
attention has been paid to universal classes of asymptotic behaviors of these
models in the limit of large particle numbers, in particular the spacings
between the particles and the fluctuation behavior of the largest particle.
Computer simulations suggest that the connections go even farther and also
comprise the zeros of the Riemann zeta function. The existing proofs require a
substantial technical machinery and heavy tools from various parts of
mathematics, in particular complex analysis, combinatorics and variational
analysis. Particularly in the last decade, a number of fine results have been
achieved, but it is obvious that a comprehensive and thorough understanding of
the matter is still lacking. Hence, it seems an appropriate time to provide a
surveying text on this research area.Comment: Published at http://dx.doi.org/10.1214/154957805100000177 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Comparison of Numerical Solvers for Differential Equations for Holonomic Gradient Method in Statistics
Definite integrals with parameters of holonomic functions satisfy holonomic
systems of linear partial differential equations. When we restrict parameters
to a one dimensional curve, the system becomes a linear ordinary differential
equation (ODE) with respect to a curve in the parameter space. We can evaluate
the integral by solving the linear ODE numerically. This approach to evaluate
numerically definite integrals is called the holonomic gradient method (HGM)
and it is useful to evaluate several normalizing constants in statistics. We
will discuss and compare methods to solve linear ODE's to evaluate normalizing
constants.Comment: 17 page
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles
We consider properties of determinants of some random symmetric matrices
issued from multivariate statistics: Wishart/Laguerre ensemble (sample
covariance matrices), Uniform Gram ensemble (sample correlation matrices) and
Jacobi ensemble (MANOVA). If is the size of the sample, the
number of variates and such a matrix, a generalization of the
Bartlett-type theorems gives a decomposition of into a product
of independent gamma or beta random variables. For fixed, we study the
evolution as grows, and then take the limit of large and with . We derive limit theorems for the sequence of {\it processes with
independent increments} for .. Since the logarithm of the determinant is a linear
statistic of the empirical spectral distribution, we connect the results for
marginals (fixed ) with those obtained by the spectral method. Actually, all
the results hold true for models, if we define the determinant as the
product of charges.Comment: 51 pages ; it replaces and extends arXiv:math/0607767 and
arXiv:math/0509021 Third version: corrected constants in Theorem 3.
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