2,690 research outputs found
Zero Lyapunov exponents of the Hodge bundle
By the results of G. Forni and of R. Trevi\~no, the Lyapunov spectrum of the
Hodge bundle over the Teichm\"uller geodesic flow on the strata of Abelian and
of quadratic differentials does not contain zeroes even though for certain
invariant submanifolds zero exponents are present in the Lyapunov spectrum. In
all previously known examples, the zero exponents correspond to those
PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy
of the Gauss-Manin connection acts by isometries of the Hodge metric. We
present an example of an arithmetic Teichm\"uller curve, for which the real
Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles,
and nevertheless its spectrum of Lyapunov exponents contains zeroes. We
describe the mechanism of this phenomenon; it covers the previously known
situation as a particular case. Conjecturally, this is the only way zero
exponents can appear in the Lyapunov spectrum of the Hodge bundle for any
PSL(2,R)-invariant probability measure.Comment: 47 pages, 10 figures. Final version (based on the referee's report).
A slightly shorter version of this article will appear in Commentarii
Mathematici Helvetici. A pdf file containing a copy of the Mathematica
routine "FMZ3-Zariski-numerics_det1.nb" is available at this link here:
http://w3.impa.br/~cmateus/files/FMZ3-Zariski-numerics_det1.pd
On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev
inequalities for a class of Boltzmann-Gibbs measures with singular interaction.
Such measures allow to model one-dimensional particles with confinement and
singular pair interaction. The functional inequalities come from convexity. We
prove and characterize optimality in the case of quadratic confinement via a
factorization of the measure. This optimality phenomenon holds for all beta
Hermite ensembles including the Gaussian unitary ensemble, a famous exactly
solvable model of random matrix theory. We further explore exact solvability by
reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting
the Hermite-Lassalle orthogonal polynomials as a complete set of
eigenfunctions. We also discuss the consequence of the log-Sobolev inequality
in terms of concentration of measure for Lipschitz functions such as maxima and
linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional
Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics
225
Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
Spectra of sparse non-Hermitian random matrices determine the dynamics of
complex processes on graphs. Eigenvalue outliers in the spectrum are of
particular interest, since they determine the stationary state and the
stability of dynamical processes. We present a general and exact theory for the
eigenvalue outliers of random matrices with a local tree structure. For
adjacency and Laplacian matrices of oriented random graphs, we derive
analytical expressions for the eigenvalue outliers, the first moments of the
distribution of eigenvector elements associated with an outlier, the support of
the spectral density, and the spectral gap. We show that these spectral
observables obey universal expressions, which hold for a broad class of
oriented random matrices.Comment: 25 pages, 4 figure
Perturbations of Dirac operators
We study general conditions under which the computations of the index of a
perturbed Dirac operator localize to the singular set of the
bundle endomorphism in the semi-classical limit . We show how
to use Witten's method to compute the index of by doing a combinatorial
computation involving local data at the nondegenerate singular points of the
operator . In particular, we provide examples of novel deformations of the
de Rham operator to establish new results relating the Euler characteristic of
a spin manifold to maps between its even and odd spinor bundles. The
paper contains a list of the current literature on the subject.Comment: 34 pages, improved results, new applications, literature list update
A Modular Sewing Kit for Entanglement Wedges
We relate the Riemann curvature of a holographic spacetime to an entanglement
property of the dual CFT state: the Berry curvature of its modular
Hamiltonians. The modular Berry connection encodes the relative bases of nearby
CFT subregions while its bulk dual, restricted to the code subspace, relates
the edge-mode frames of the corresponding entanglement wedges. At leading order
in 1/N and for sufficiently smooth HRRT surfaces, the modular Berry connection
simply sews together the orthonormal coordinate systems covering neighborhoods
of HRRT surfaces. This geometric perspective on entanglement is a promising new
tool for connecting the dynamics of entanglement and gravitation.Comment: 26 pages + Appendices, 4 figure
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