4,127 research outputs found
Topological recursion for Masur-Veech volumes
We study the Masur-Veech volumes of the principal stratum of the
moduli space of quadratic differentials of unit area on curves of genus
with punctures. We show that the volumes are the constant terms
of a family of polynomials in variables governed by the topological
recursion/Virasoro constraints. This is equivalent to a formula giving these
polynomials as a sum over stable graphs, and retrieves a result of
\cite{Delecroix} proved by combinatorial arguments. Our method is different: it
relies on the geometric recursion and its application to statistics of
hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain
an expression of the area Siegel--Veech constants in terms of hyperbolic
geometry. The topological recursion allows numerical computations of
Masur--Veech volumes, and thus of area Siegel--Veech constants, for low and
, which leads us to propose conjectural formulas for low but all . We
also relate our polynomials to the asymptotic counting of square-tiled surfaces
with large boundaries.Comment: 75 pages, v2: added a section on enumeration of square-tiled surface
Quadratic maps with a periodic critical point of period 2
We provide a complete classification of possible graphs of rational
preperiodic points of endomorphisms of the projective line of degree 2 defined
over the rationals with a rational periodic critical point of period 2, under
the assumption that these maps have no periodic points of period at least 7. We
explain how this extends results of Poonen on quadratic polynomials. We show
that there are 13 possible graphs, and that such maps have at most 9 rational
preperiodic points. We provide data related to the analogous classification of
graphs of endomorphisms of degree 2 with a rational periodic critical point of
period 3 or 4.Comment: Updated theorem 2 to rule out the cases of quadratic maps with a
rational periodic critical point of period 2 and a rational periodic point of
period 5 or
Tropical Hurwitz Numbers
Hurwitz numbers count genus g, degree d covers of the projective line with
fixed branch locus. This equals the degree of a natural branch map defined on
the Hurwitz space. In tropical geometry, algebraic curves are replaced by
certain piece-wise linear objects called tropical curves. This paper develops a
tropical counterpart of the branch map and shows that its degree recovers
classical Hurwitz numbers.Comment: Published in Journal of Algebraic Combinatorics, Volume 32, Number 2
/ September, 2010. Added section on genus zero piecewise polynomiality.
Removed paragraph on psi classe
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
Shapes of interacting RNA complexes
Shapes of interacting RNA complexes are studied using a filtration via their
topological genus. A shape of an RNA complex is obtained by (iteratively)
collapsing stacks and eliminating hairpin loops. This shape-projection
preserves the topological core of the RNA complex and for fixed topological
genus there are only finitely many such shapes.Our main result is a new
bijection that relates the shapes of RNA complexes with shapes of RNA
structures.This allows to compute the shape polynomial of RNA complexes via the
shape polynomial of RNA structures. We furthermore present a linear time
uniform sampling algorithm for shapes of RNA complexes of fixed topological
genus.Comment: 38 pages 24 figure
Exact 2-point function in Hermitian matrix model
J. Harer and D. Zagier have found a strikingly simple generating function for
exact (all-genera) 1-point correlators in the Gaussian Hermitian matrix model.
In this paper we generalize their result to 2-point correlators, using Toda
integrability of the model. Remarkably, this exact 2-point correlation function
turns out to be an elementary function - arctangent. Relation to the standard
2-point resolvents is pointed out. Some attempts of generalization to 3-point
and higher functions are described.Comment: 31 pages, 1 figur
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