45,457 research outputs found

### Fully Packed O(n=1) Model on Random Eulerian Triangulations

We introduce a matrix model describing the fully-packed O(n) model on random
Eulerian triangulations (i.e. triangulations with all vertices of even
valency). For n=1 the model is mapped onto a particular gravitational 6-vertex
model with central charge c=1, hence displaying the expected shift c -> c+1
when going from ordinary random triangulations to Eulerian ones. The case of
arbitrary n is also discussed.Comment: 12 pages, 3 figures, tex, harvmac, eps

### SU(N) Meander Determinants

We propose a generalization of meanders, i.e., configurations of
non-selfintersecting loops crossing a line through a given number of points, to
SU(N). This uses the reformulation of meanders as pairs of reduced elements of
the Temperley-Lieb algebra, a SU(2)-related quotient of the Hecke algebra, with
a natural generalization to SU(N). We also derive explicit formulas for SU(N)
meander determinants, defined as the Gram determinants of the corresponding
bases of the Hecke algebra.Comment: TeX using harvmac.tex and epsf.tex, 60 pages (l-mode), 5 figure

### Noncommutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete
integrable equations, notably $Q$ and $T$-systems, is given in terms of
partition functions of positively weighted paths, thereby proving the positive
Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of
solution is amenable to generalization to non-commutative weighted paths. Under
certain circumstances, these describe solutions of discrete evolution equations
in non-commutative variables: Examples are the corresponding quantum cluster
algebras [BZ], the Kontsevich evolution [DFK09b] and the $T$-systems themselves
[DFK09a]. In this paper, we formulate certain non-commutative integrable
evolutions by considering paths with non-commutative weights, together with an
evolution of the weights that reduces to cluster algebra mutations in the
commutative limit. The general weights are expressed as Laurent monomials of
quasi-determinants of path partition functions, allowing for a non-commutative
version of the positive Laurent phenomenon. We apply this construction to the
known systems, and obtain Laurent positivity results for their solutions in
terms of initial data.Comment: 46 pages, minor typos correcte

### Coloring Random Triangulations

We introduce and solve a two-matrix model for the tri-coloring problem of the
vertices of a random triangulation. We present three different solutions: (i)
by orthogonal polynomial techniques (ii) by use of a discrete Hirota bilinear
equation (iii) by direct expansion. The model is found to lie in the
universality class of pure two-dimensional quantum gravity, despite the
non-polynomiality of its potential.Comment: 50 pages, 4 figures, Tex, uses harvmac, eps

### Combinatorial point for higher spin loop models

Integrable loop models associated with higher representations (spin k/2) of
U_q(sl(2)) are investigated at the point q=-e^{i\pi/(k+2)}. The ground state
eigenvalue and eigenvectors are described. Introducing inhomogeneities into the
models allows to derive a sum rule for the ground state entries.Comment: latest version adds some reference

### Folding Transitions of the Square-Diagonal Lattice

We address the problem of "phantom" folding of the tethered membrane modelled
by the two-dimensional square lattice, with bonds on the edges and diagonals of
each face. Introducing bending rigidities $K_1$ and $K_2$ for respectively long
and short bonds, we derive the complete phase diagram of the model, using
transfer matrix calculations. The latter displays two transition curves, one
corresponding to a first order (ferromagnetic) folding transition, and the
other to a continuous (anti-ferromagnetic) unfolding transition.Comment: TeX using harvmac.tex and epsf.tex, 22 pages (l mode), 17 figure

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