2,472 research outputs found
A Lichnerowicz estimate for the spectral gap of the sub-Laplacian
For a second order operator on a compact manifold satisfying the strong
H\"ormander condition, we give a bound for the spectral gap analogous to the
Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a
wide class of such operators which includes horizontal lifts of the Laplacian
on Riemannian submersions with minimal leaves.Comment: 13 pages. To appear in Proceedings of the AM
Heights, Regulators and Schinzel's determinant inequality
We prove inequalities that compare the size of an S-regulator with a product
of heights of multiplicatively independent S-units. Our upper bound for the
S-regulator follows from a general upper bound for the determinant of a real
matrix proved by Schinzel. The lower bound for the S-regulator follows from
Minkowski's theorem on successive minima and a volume formula proved by Meyer
and Pajor. We establish similar upper bounds for the relative regulator of an
extension of number fields.Comment: accepted for Publication in Acta Arit
Tracy-Widom GUE law and symplectic invariants
We establish the relation between two objects: an integrable system related
to Painleve II equation, and the symplectic invariants of a certain plane curve
\Sigma_{TW} describing the average eigenvalue density of a random hermitian
matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This
explains directly how the Tracy-Widow law F_{GUE}, governing the distribution
of the maximal eigenvalue in hermitian random matrices, can also be recovered
from symplectic invariants.Comment: pdfLatex, 36 pages, 1 figure. v2: typos corrected, re-sectioning, a
reference adde
Loop equations from differential systems
To any differential system where belongs to a Lie
group (a fiber of a principal bundle) and is a Lie algebra
valued 1-form on a Riemann surface , is associated an infinite sequence
of "correlators" that are symmetric -forms on . The goal of
this article is to prove that these correlators always satisfy "loop
equations", the same equations satisfied by correlation functions in random
matrix models, or the same equations as Virasoro or W-algebra constraints in
CFT.Comment: 20 page
Reversible electrowetting and trapping of charge: model and experiments
We derive a model for voltage-induced wetting, so-called electrowetting, from
the principle of virtual displacement. Our model includes the possibility that
charge is trapped in or on the wetted surface. Experimentally, we show
reversible electrowetting for an aqueous droplet on an insulating layer of 10
micrometer thickness. The insulator is coated with a highly fluorinated layer
impregnated with oil, providing a contact-angle hysteresis lower than 2
degrees. Analyzing the data with our model, we find that until a threshold
voltage of 240 V, the induced charge remains in the liquid and is not trapped.
For potentials beyond the threshold, the wetting force and the contact angle
saturate, in line with the occurrence of trapping of charge in or on the
insulating layer. The data are independent of the polarity of the applied
electric field, and of the ion type and molarity. We suggest possible
microscopic origins for charge trapping.Comment: 13 pages & 5 figures; the paper has been accepted for publication in
Langmui
The sine-law gap probability, Painlev\'e 5, and asymptotic expansion by the topological recursion
The goal of this article is to rederive the connection between the Painlev\'e
integrable system and the universal eigenvalues correlation functions of
double-scaled hermitian matrix models, through the topological recursion
method. More specifically we prove, \textbf{to all orders}, that the WKB
asymptotic expansions of the -function as well as of determinantal
formulas arising from the Painlev\'e Lax pair are identical to the large
double scaling asymptotic expansions of the partition function and
correlation functions of any hermitian matrix model around a regular point in
the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic
of large random matrices and provide an alternative perturbative proof of
universality in the bulk with only algebraic methods. Eventually we exhibit the
first orders of the series expansion up to .Comment: 37 pages, 1 figure, published in Random Matrices: Theory and
Application
Average characteristic polynomials in the two-matrix model
The two-matrix model is defined on pairs of Hermitian matrices of
size by the probability measure where
and are given potential functions and \tau\in\er. We study averages
of products and ratios of characteristic polynomials in the two-matrix model,
where both matrices and may appear in a combined way in both
numerator and denominator. We obtain determinantal expressions for such
averages. The determinants are constructed from several building blocks: the
biorthogonal polynomials and associated to the two-matrix
model; certain transformed functions and \Q_n(v); and finally
Cauchy-type transforms of the four Eynard-Mehta kernels , ,
and . In this way we generalize known results for the
-matrix model. Our results also imply a new proof of the Eynard-Mehta
theorem for correlation functions in the two-matrix model, and they lead to a
generating function for averages of products of traces.Comment: 28 pages, references adde
WKB solutions of difference equations and reconstruction by the topological recursion
The purpose of this article is to analyze the connection between
Eynard-Orantin topological recursion and formal WKB solutions of a
-difference equation: with .
In particular, we extend the notion of determinantal formulas and topological
type property proposed for formal WKB solutions of -differential systems
to this setting. We apply our results to a specific -difference system
associated to the quantum curve of the Gromov-Witten invariants of
for which we are able to prove that the correlation functions
are reconstructed from the Eynard-Orantin differentials computed from the
topological recursion applied to the spectral curve .
Finally, identifying the large expansion of the correlation functions,
proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new
generating series for Gromov-Witten invariants of .Comment: 41 pages, 2 figures, published version in Nonlinearit
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