110 research outputs found
A Solvable 2D Quantum Gravity Model with \GAMMA >0
We consider a model of discretized 2d gravity interacting with Ising spins
where phase boundaries are restricted to have minimal length and show
analytically that the critical exponent at the spin transition
point. The model captures the numerically observed behavior of standard
multiple Ising spins coupled to 2d gravity.Comment: Latex, 9 pages, NBI-HE-94-0
A restricted dimer model on a 2-dimensional random causal triangulation
We introduce a restricted hard dimer model on a random causal triangulation
that is exactly solvable and generalizes a model recently proposed by Atkin and
Zohren. We show that the latter model exhibits unusual behaviour at its
multicritical point; in particular, its Hausdorff dimension equals 3 and not
3/2 as would be expected from general scaling arguments. When viewed as a
special case of the generalized model introduced here we show that this
behaviour is not generic and therefore is not likely to represent the true
behaviour of the full dimer model on a random causal triangulation.Comment: 26 pages, typos corrected, slight generalization adde
Universality of hypercubic random surfaces
We study universality properties of the Weingarten hyper-cubic random
surfaces. Since a long time ago the model with a local restriction forbidding
surface self-bendings has been thought to be in a different universality class
from the unrestricted model defined on the full set of surfaces. We show that
both models in fact belong to the same universality class with the entropy
exponent gamma = 1/2 and differ by finite size effects which are much more
pronounced in the restricted model.Comment: 8 pages, 3 figure
Topological quantum field theory and invariants of graphs for quantum groups
On basis of generalized 6j-symbols we give a formulation of topological
quantum field theories for 3-manifolds including observables in the form of
coloured graphs. It is shown that the 6j-symbols associated with deformations
of the classical groups at simple even roots of unity provide examples of this
construction. Calculational methods are developed which, in particular, yield
the dimensions of the state spaces as well as a proof of the relation,
previously announced for the case of by V.Turaev, between these
models and corresponding ones based on the ribbon graph construction of
Reshetikhin and Turaev.Comment: 38 page
On the relation between two quantum group invariants of 3-cobordisms
We prove in the context of quantum groups at even roots of unity that a Turaev-Viro type invariant of a three-dimensional cobordism M equals the tensor product of the Reshetikhin-Turaev invariants of M and M*, where the latter denotes M with orientation reversed
The spectral dimension of generic trees
We define generic ensembles of infinite trees. These are limits as
of ensembles of finite trees of fixed size , defined in terms
of a set of branching weights. Among these ensembles are those supported on
trees with vertices of a uniformly bounded order. The associated probability
measures are supported on trees with a single spine and Hausdorff dimension
. Our main result is that their spectral dimension is , and
that the critical exponent of the mass, defined as the exponential decay rate
of the two-point function along the spine, is 1/3
Condensation in nongeneric trees
We study nongeneric planar trees and prove the existence of a Gibbs measure
on infinite trees obtained as a weak limit of the finite volume measures. It is
shown that in the infinite volume limit there arises exactly one vertex of
infinite degree and the rest of the tree is distributed like a subcritical
Galton-Watson tree with mean offspring probability . We calculate the rate
of divergence of the degree of the highest order vertex of finite trees in the
thermodynamic limit and show it goes like where is the size of the
tree. These trees have infinite spectral dimension with probability one but the
spectral dimension calculated from the ensemble average of the generating
function for return probabilities is given by if the weight
of a vertex of degree is asymptotic to .Comment: 57 pages, 14 figures. Minor change
Random trees with superexponential branching weights
We study rooted planar random trees with a probability distribution which is
proportional to a product of weight factors associated to the vertices of
the tree and depending only on their individual degrees . We focus on the
case when grows faster than exponentially with . In this case the
measures on trees of finite size converge weakly as tends to infinity
to a measure which is concentrated on a single tree with one vertex of infinite
degree. For explicit weight factors of the form with
we obtain more refined results about the approach to the infinite
volume limit.Comment: 19 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
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