85 research outputs found
Expoential bounds on the number of causal triangulations
We prove that the number of combinatorially distinct causal 3-dimensional
triangulations homeomorphic to the 3-dimensional sphere is bounded by an
exponential function of the number of tetrahedra. It is also proven that the
number of combinatorially distinct causal 4-dimensional triangulations
homeomorphic to the 4-sphere is bounded by an exponential function of the
number of 4-simplices provided the number of all combinatorially distinct
triangulations of the 3-sphere is bounded by an exponential function of the
number of tetrahedra.Comment: 30 pages, 9 figure
Remarks on the entropy of 3-manifolds
We give a simple combinatoric proof of an exponential upper bound on the
number of distinct 3-manifolds that can be constructed by successively
identifying nearest neighbour pairs of triangles in the boundary of a
simplicial 3-ball and show that all closed simplicial manifolds that can be
constructed in this manner are homeomorphic to . We discuss the problem of
proving that all 3-dimensional simplicial spheres can be obtained by this
construction and give an example of a simplicial 3-ball whose boundary
triangles can be identified pairwise such that no triangle is identified with
any of its neighbours and the resulting 3-dimensional simplicial complex is a
simply connected 3-manifold.Comment: 12 pages, 5 figures available from author
A polymer gas on a random surface
Using the observation that configurations of N polymers with hard core
interactions on a closed random surface correspond to random surfaces with N
boundary components we calculate the free energy of a gas of polymers
interacting with fully quantized two-dimensional gravity. We derive the
equation of state for the polymer gas and find that all the virial coefficients
beyond the second one vanish identically.Comment: 6 page
Local limit of labeled trees and expected volume growth in a random quadrangulation
Exploiting a bijective correspondence between planar quadrangulations and
well-labeled trees, we define an ensemble of infinite surfaces as a limit of
uniformly distributed ensembles of quadrangulations of fixed finite volume. The
limit random surface can be described in terms of a birth and death process and
a sequence of multitype Galton--Watson trees. As a consequence, we find that
the expected volume of the ball of radius around a marked point in the
limit random surface is .Comment: Published at http://dx.doi.org/10.1214/009117905000000774 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Branched polymers on branched polymers
We study an ensemble of branched polymers which are embedded on other
branched polymers. This is a toy model which allows us to study explicitly the
reaction of a statistical system on an underlying geometrical structure, a
problem of interest in the study of the interaction of matter and quantized
gravity. We find a phase transition at which the embedded polymers begin to
cover the basis polymers. At the phase transition point the susceptibility
exponent takes the value 3/4 and the two-point function develops an
anomalous dimension 1/2.Comment: uuencoded 9 p. ps-file + 2 ps-figure
Classification and construction of unitary topological field theories in two dimensions
We prove that unitary two-dimensional topological field theories are uniquely
characterized by positive real numbers which
can be regarded as the eigenvalues of a hermitean handle creation operator. The
number is the dimension of the Hilbert space associated with the circle and
the partition functions for closed surfaces have the form where is the genus. The eigenvalues
can be arbitary positive numbers. We show how such a theory can be constructed
on triangulated surfaces.Comment: 12 pages, late
On the entropy of LEGO
We propose the further study of the rate of growth of the number of
contiguous buildings which may be made from n LEGO blocks of the same size and
color. Specializing to blocks of dimension 2x4 we give upper and lower bounds,
and speculate on the true value.Comment: 13 pages, 7 figures. Revised version: Minor corrections, page
Generic Ising Trees
The Ising model on an infinite generic tree is defined as a thermodynamic
limit of finite systems. A detailed description of the corresponding
distribution of infinite spin configurations is given. As an application we
study the magnetization properties of such systems and prove that they exhibit
no spontaneous magnetization. Furthermore, the values of the Hausdorff and
spectral dimensions of the underlying trees are calculated and found to be,
respectively, and .Comment: 29 pages, 2 figures; typos corrected, one section and new references
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The Existence and Stability of Noncommutative Scalar Solitons
We establish existence and stabilty results for solitons in noncommutative
scalar field theories in even space dimension . In particular, for any
finite rank spectral projection of the number operator of
the -dimensional harmonic oscillator and sufficiently large noncommutativity
parameter we prove the existence of a rotationally invariant soliton
which depends smoothly on and converges to a multiple of as
.
In the two-dimensional case we prove that these solitons are stable at large
, if , where projects onto the space spanned by the
lowest eigenstates of , and otherwise they are unstable. We also
discuss the generalisation of the stability results to higher dimensions. In
particular, we prove stability of the soliton corresponding to for all
in its domain of existence.
Finally, for arbitrary and small values of , we prove without
assuming rotational invariance that there do not exist any solitons depending
smoothly on .Comment: 36 pages, 1 figur
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