10,709 research outputs found
Curvature Matrix Models for Dynamical Triangulations and the Itzykson-DiFrancesco Formula
We study the large-N limit of a class of matrix models for dually weighted
triangulated random surfaces using character expansion techniques. We show that
for various choices of the weights of vertices of the dynamical triangulation
the model can be solved by resumming the Itzykson-Di Francesco formula over
congruence classes of Young tableau weights modulo three. From this we show
that the large-N limit implies a non-trivial correspondence with models of
random surfaces weighted with only even coordination number vertices. We
examine the critical behaviour and evaluation of observables and discuss their
interrelationships in all models. We obtain explicit solutions of the model for
simple choices of vertex weightings and use them to show how the matrix model
reproduces features of the random surface sum. We also discuss some general
properties of the large-N character expansion approach as well as potential
physical applications of our results.Comment: 37 pages LaTeX; Some clarifying comments added, last Section
rewritte
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 Critical Point for Two Dimensional Holomorphics Systems
Let be a biholomorphisms on two--dimensional a complex
manifold, and let be a compact --invariant set such that
is asymptotically dissipative and without sinks periodic points. We
introduce a solely dynamical obstruction to dominated splitting, namely
critical point. Critical point is a dynamical object and capture many of the
dynamical properties of their one--dimensional counterpart
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