4,287 research outputs found
Asymptotics of skew orthogonal polynomials
Exact integral expressions of the skew orthogonal polynomials involved in
Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are
obtained: the (even rank) skew orthogonal polynomials are average
characteristic polynomials of random matrices. From there, asymptotics of the
skew orthogonal polynomials are derived.Comment: 17 pages, Late
Representations of classical Lie groups and quantized free convolution
We study the decompositions into irreducible components of tensor products
and restrictions of irreducible representations of classical Lie groups as the
rank of the group goes to infinity. We prove the Law of Large Numbers for the
random counting measures describing the decomposition. This leads to two
operations on measures which are deformations of the notions of the free
convolution and the free projection. We further prove that if one replaces
counting measures with others coming from the work of Perelomov and Popov on
the higher order Casimir operators for classical groups, then the operations on
the measures turn into the free convolution and projection themselves.
We also explain the relation between our results and limit shape theorems for
uniformly random lozenge tilings with and without axial symmetry.Comment: 43 pages, 4 figures. v3: relation to the Markov-Krein correspondence
is updated and correcte
Probing thermality beyond the diagonal
We investigate the off-diagonal sector of eigenstate thermalization using
both local and non-local probes in 2-dimensional conformal field theories. A
novel analysis of the asymptotics of OPE coefficients via the modular bootstrap
is performed to extract the behaviour of the off-diagonal matrix elements. We
also probe this sector using semi-classical heavy-light Virasoro blocks. The
results demonstrate signatures of thermality and confirms the entropic
suppression of the off-diagonal elements as necessitated by the eigenstate
thermalization hypothesis.Comment: 27 pages, 2 figure
High Temperature Asymptotics of Orthogonal Mean-Field Spin Glasses
We evaluate the high temperature limit of the free energy of spin glasses on
the hypercube with Hamiltonian , where the
coupling matrix is drawn from certain symmetric orthogonally invariant
ensembles. Our derivation relates the annealed free energy of these models to a
spherical integral, and expresses the limit of the free energy in terms of the
limiting spectral measure of the coupling matrix . As an application, we
derive the limiting free energy of the Random Orthogonal Model (ROM) at high
temperatures, which confirms non-rigorous calculations of Marinari et al.
(1994). Our methods also apply to other well-known models of disordered
systems, including the SK and Gaussian Hopfield models.Comment: 15 pages, 1 figur
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail
We prove precise deviations results in the sense of Cram\'er and Petrov for
the upper tail of the distribution of the maximal value for a special class of
determinantal point processes that play an important role in random matrix
theory. Here we cover all three regimes of moderate, large and superlarge
deviations for which we determine the leading order description of the tail
probabilities. As a corollary of our results we identify the region within the
regime of moderate deviations for which the limiting Tracy-Widom law still
predicts the correct leading order behavior. Our proofs use that the
determinantal point process is given by the Christoffel-Darboux kernel for an
associated family of orthogonal polynomials. The necessary asymptotic
information on this kernel has mostly been obtained in [Kriecherbauer T.,
Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015),
639-694]. In the superlarge regime these results of do not suffice and we put
stronger assumptions on the point processes. The results of the present paper
and the relevant parts of [Kriecherbauer T., Schubert K., Sch\"uler K., Venker
M., Markov Process. Related Fields 21 (2015), 639-694] have been proved in the
dissertation [Sch\"uler K., Ph.D. Thesis, Universit\"at Bayreuth, 2015].Comment: 18 page
A proposal of a methodological framework with experimental guidelines to investigate clustering stability on financial time series
We present in this paper an empirical framework motivated by the practitioner
point of view on stability. The goal is to both assess clustering validity and
yield market insights by providing through the data perturbations we propose a
multi-view of the assets' clustering behaviour. The perturbation framework is
illustrated on an extensive credit default swap time series database available
online at www.datagrapple.com.Comment: Accepted at ICMLA 201
- …