19,157 research outputs found
A New Recursion Relation for the 6j-Symbol
The 6j-symbol is a fundamental object from the re-coupling theory of SU(2)
representations. In the limit of large angular momenta, its asymptotics is
known to be described by the geometry of a tetrahedron with quantized lengths.
This article presents a new recursion formula for the square of the 6j-symbol.
In the asymptotic regime, the new recursion is shown to characterize the
closure of the relevant tetrahedron. Since the 6j-symbol is the basic building
block of the Ponzano-Regge model for pure three-dimensional quantum gravity, we
also discuss how to generalize the method to derive more general recursion
relations on the full amplitudes.Comment: 10 pages, v2: title and introduction changed, paper re-structured;
Annales Henri Poincare (2011
On ELSV-type formulae, Hurwitz numbers and topological recursion
We present several recent developments on ELSV-type formulae and topological
recursion concerning Chiodo classes and several kind of Hurwitz numbers. The
main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine,
"Chiodo formulas for the r-th roots and topological recursion", Lett. Math.
Phys. (2016).Comment: 18 pages, comments are welcom
Chiodo formulas for the r-th roots and topological recursion
We analyze Chiodo's formulas for the Chern classes related to the r-th roots
of the suitably twisted integer powers of the canonical class on the moduli
space of curves. The intersection numbers of these classes with psi-classes are
reproduced via the Chekhov-Eynard-Orantin topological recursion. As an
application, we prove that the Johnson-Pandharipande-Tseng formula for the
orbifold Hurwitz numbers is equivalent to the topological recursion for the
orbifold Hurwitz numbers. In particular, this gives a new proof of the
topological recursion for the orbifold Hurwitz numbers.Comment: 19 pages, some correction
Recursion Operators and Frobenius Manifolds
In this note I exhibit a "discrete homotopy" which joins the category of
F-manifolds to the category of Poisson-Nijenhuis manifolds, passing through the
category of Frobenius manifolds
Non-homogenous disks in the chain of matrices
We investigate the generating functions of multi-colored discrete disks with
non-homogenous boundary conditions in the context of the Hermitian multi-matrix
model where the matrices are coupled in an open chain. We show that the study
of the spectral curve of the matrix model allows one to solve a set of loop
equations to get a recursive formula computing mixed trace correlation
functions to leading order in the large matrix limit.Comment: 25 pages, 4 figure
Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
For Anderson localization on the Cayley tree, we study the statistics of
various observables as a function of the disorder strength and the number
of generations. We first consider the Landauer transmission . In the
localized phase, its logarithm follows the traveling wave form where (i) the disorder-averaged value moves linearly
and the localization length
diverges as with (ii) the
variable is a fixed random variable with a power-law tail for large with , so that all
integer moments of are governed by rare events. In the delocalized phase,
the transmission remains a finite random variable as , and
we measure near criticality the essential singularity with . We then consider the
statistical properties of normalized eigenstates, in particular the entropy and
the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical
entropy diverges as with , whereas it grows
linearly in in the delocalized phase. Finally for the I.P.R., we explain
how closely related variables propagate as traveling waves in the delocalized
phase. In conclusion, both the localized phase and the delocalized phase are
characterized by the traveling wave propagation of some probability
distributions, and the Anderson localization/delocalization transition then
corresponds to a traveling/non-traveling critical point. Moreover, our results
point towards the existence of several exponents at criticality.Comment: 28 pages, 21 figures, comments welcom
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