On the width of handles in two-dimensional quantum gravity

We discuss the average length l of the shortest non-contractible loop on surfaces in the two-dimensional pure quantum gravity ensemble. The value of $\gamma_{str}$ and the explicit form of the loop functions indicate that l diverges at the critical point. Scaling arguments suggest that the critical exponent of l is 1/2. We show that this value of the critical exponent is also obtained for branched polymers where the calculation is straightforward.Comment: 7 pages, 1 ps figure, late

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

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

Classification and construction of unitary topological field theories in two dimensions

We prove that unitary two-dimensional topological field theories are uniquely characterized by $n$ positive real numbers $\lambda _1,\ldots \lambda _n$ which can be regarded as the eigenvalues of a hermitean handle creation operator. The number $n$ is the dimension of the Hilbert space associated with the circle and the partition functions for closed surfaces have the form $$Z_g=\sum_{i=1}^{n}\lambda _i^{g-1}$$ where $g$ 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

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 $S^3$. 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

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 $\gamma$ 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

The spectral dimension of the branched polymers phase of two-dimensional quantum gravity

The metric of two-dimensional quantum gravity interacting with conformal matter is believed to collapse to a branched polymer metric when the central charge c>1. We show analytically that the spectral dimension of such a branched polymer phase is four thirds. This is in good agreement with numerical simulations for large c.Comment: 29 pages plain LateX2e, 7 eps figures included using eps

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 $2d$. In particular, for any finite rank spectral projection $P$ of the number operator ${\mathcal N}$ of the $d$-dimensional harmonic oscillator and sufficiently large noncommutativity parameter $\theta$ we prove the existence of a rotationally invariant soliton which depends smoothly on $\theta$ and converges to a multiple of $P$ as $\theta\to\infty$. In the two-dimensional case we prove that these solitons are stable at large $\theta$, if $P=P_N$, where $P_N$ projects onto the space spanned by the $N+1$ lowest eigenstates of ${\mathcal N}$, 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 $P=P_0$ for all $\theta$ in its domain of existence. Finally, for arbitrary $d$ and small values of $\theta$, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on $\theta$.Comment: 36 pages, 1 figur