713 research outputs found
More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2
For the model on a random lattice has critical points to
which a scaling behaviour characteristic of 2D gravity interacting with
conformal matter fields with can be associated. Previously
we have written down an exact solution of this model valid at any point in the
coupling constant space and for any . The solution was parametrized in terms
of an auxiliary function. Here we determine the auxiliary function explicitly
as a combination of -functions, thereby completing the solution of the
model. Using our solution we investigate, for the simplest version of the
model, hitherto unexplored regions of the parameter space. For example we
determine in a closed form the eigenvalue density without any assumption of
being close to or at a critical point. This gives a generalization of the
Wigner semi-circle law to . We also study the model for . Both
for we find that the model is well defined in a certain region
of the coupling constant space. For we find no new critical points while
for we find new critical points at which the string susceptibility
exponent takes the value .Comment: 27 pages, LaTeX file (uses epsf) + 3 eps figures, formulas involving
the string susceptibility corrrected, no change in conclusion
Quasiperiodic Solutions of the Fibre Optics Coupled Nonlinear Schr{\"o}dinger Equations
We consider travelling periodical and quasiperiodical waves in single mode
fibres, with weak birefringence and under the action of cross-phase modulation.
The problem is reduced to the ``1:2:1" integrable case of the two-particle
quartic potential. A general approach for finding elliptic solutions is given.
New solutions which are associated with two-gap Treibich-Verdier potentials are
found. General quasiperiodic solutions are given in terms of two dimensional
theta functions with explicit expressions for frequencies in terms of theta
constants. The reduction of quasiperiodic solutions to elliptic functions is
discussed.Comment: 24 page
Phase Structure of the O(n) Model on a Random Lattice for n>2
We show that coarse graining arguments invented for the analysis of
multi-spin systems on a randomly triangulated surface apply also to the O(n)
model on a random lattice. These arguments imply that if the model has a
critical point with diverging string susceptibility, then either \g=+1/2 or
there exists a dual critical point with negative string susceptibility
exponent, \g', related to \g by \g=\g'/(\g'-1). Exploiting the exact solution
of the O(n) model on a random lattice we show that both situations are realized
for n>2 and that the possible dual pairs of string susceptibility exponents are
given by (\g',\g)=(-1/m,1/(m+1)), m=2,3,.... We also show that at the critical
points with positive string susceptibility exponent the average number of loops
on the surface diverges while the average length of a single loop stays finite.Comment: 18 pages, LaTeX file, two eps-figure
An Iterative Solution of the Three-colour Problem on a Random Lattice
We study the generalisation of Baxter's three-colour problem to a random
lattice. Rephrasing the problem as a matrix model problem we discuss the
analyticity structure and the critical behaviour of the resulting matrix model.
Based on a set of loop equations we develop an algorithm which enables us to
solve the three-colour problem recursivelyComment: 14 pages, LaTeX, misprints corrected, approximation of (6.20) refine
Hermitian Matrix Model with Plaquette Interaction
We study a hermitian -matrix model with plaquette interaction,
. By means of a conformal transformation we rewrite the
model as an model on a random lattice with a non polynomial potential.
This allows us to solve the model exactly. We investigate the critical
properties of the plaquette model and find that for the model
belongs to the same universality class as the model on a random lattice.Comment: 15 pages, no figures, two references adde
Boundary operators in the O(n) and RSOS matrix models
We study the new boundary condition of the O(n) model proposed by Jacobsen
and Saleur using the matrix model. The spectrum of boundary operators and their
conformal weights are obtained by solving the loop equations. Using the
diagrammatic expansion of the matrix model as well as the loop equations, we
make an explicit correspondence between the new boundary condition of the O(n)
model and the "alternating height" boundary conditions in RSOS model.Comment: 29 pages, 4 figures; version to appear in JHE
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