Frustrating and Diluting Dynamical Lattice Ising Spins

We investigate what happens to the third order ferromagnetic phase transition displayed by the Ising model on various dynamical planar lattices (ie coupled to 2D quantum gravity) when we introduce annealed bond disorder in the form of either antiferromagnetic couplings or null couplings. We also look at the effect of such disordering for the Ising model on general $\phi^3$ and $\phi^4$ Feynman diagrams.Comment: 7pages, LaTex , LPTHE-ORSAY-94-5

Is There Quantum Gravity in Two Dimensions?

A hybrid model which allows to interpolate between the (original) Regge approach and dynamical triangulations is introduced. The gained flexibility in the measure is exploited to study dynamical triangulation in a fixed geometry. Our numerical results support KPZ exponents. A critical assessment concerning the apparent lack of gravitational effects in two dimensions follows.Comment: 20 pages including 4 figures, uuencoded Z-compressed .tar file created by uufile

Ising (anti-)ferromagnet on dynamical triangulations and quadrangulations

We write down matrix models for Ising spins with zero external field on the vertices of dynamical triangulated random surfaces (DTRS) and dynamically quadrangulated random surfaces (DQRS) and compare these with the standard matrix model approach which places the spins on the dual $\phi^3$ and $\phi^4$ graphs. We show that the critical temperatures calculated in the DTRS and DQRS models agree with those deduced from duality arguments in the standard approach. Using the DQRS model we observe that the Ising antiferromagnet still undergoes a phase transition to a Neel (checkerboard) ordered ground state which is absent because of frustration in the other cases.Comment: 5 pages, late

Measuring the string susceptibility in 2D simplicial quantum gravity using the Regge approach

We use Monte Carlo simulations to study pure 2D Euclidean quantum gravity with $R^2$-interaction on spherical topologies, employing Regge's formulation. We attempt to measure the string susceptibility exponent $\gamma_{\rm str}$ by using a finite-size scaling Ansatz in the expectation value of $R^2$, as has been done in a previous study by Bock and Vink ( hep-lat/9406018 ). By considerably extending the range and statistics of their study we find that this Ansatz is plagued by large systematic errors. The $R^2$ specific string susceptibility exponent \GS' is found to agree with theoretical predictions, but its determination also is subject to large systematic errors and the presence of finite-size scaling corrections. To circumvent this obstacle we suggest a new scaling Ansatz which in principle should be able to predict both, \GS and \GS'. First results indicate that this requires large system sizes to reduce the uncertainties in the finite-size scaling Ans\"atze. Nevertheless, our investigation shows that within the achievable accuracy the numerical estimates are still compatible with analytic predictions, contrary to the recent claim by Bock and Vink.Comment: 33 pages, self unpacking uuencoded PostScript file, including all the figures. Paper also available at http://www.physik.fu-berlin.de/~holm

Square Gravity

We simulate the Ising model on dynamical quadrangulations using a generalization of the flip move for triangulations with two aims: firstly, as a confirmation of the universality of the KPZ/DDK exponents of the Ising phase transition, worthwhile in view of some recent surprises with other sorts of dynamical lattices; secondly, to investigate the transition of the Ising antiferromagnet on a dynamical loosely packed (bipartite) lattice. In the latter case we show that it is still possible to define a staggered magnetization and observe the antiferromagnetic analogue of the transition.Comment: LaTeX file and 7 postscript figures bundled together with uufile

Multiple Potts Models Coupled to Two-Dimensional Quantum Gravity

We perform Monte Carlo simulations using the Wolff cluster algorithm of {\it multiple} $q=2,3,4$ state Potts models on dynamical phi-cubed graphs of spherical topology in order to investigate the $c>1$ region of two-dimensional quantum gravity. Contrary to naive expectation we find no obvious signs of pathological behaviour for $c>1$. We discuss the results in the light of suggestions that have been made for a modified DDK ansatz for $c>1$.Comment: 9 page

Toward an analytic determination of the deconfinement temperature in SU(2) L.G.T.

We consider the SU(2) lattice gauge theory at finite temperature in (d+1) dimensions, with different couplings $\beta_t$ and $\beta_s$ for timelike and spacelike plaquettes. By using the character expansion of the Wilson action and performing the integrals over space-like link variables, we find an effective action for the Polyakov loops which is exact to all orders in $\beta_t$ and to the first non-trivial order in $\beta_s$. The critical coupling for the deconfinement transition is determined in the (3+1) dimensional case, by the mean field method, for different values of the lattice size $N_t$ in the compactified time direction and of the asymmetry parameter $\rho = \sqrt{\beta_t/\beta_s}$. We find good agreement with Montecarlo simulations in the range $1\leq N_t \leq 5$, and good qualitative agreement in the same range with the logarithmic scaling law of QCD. Moreover the dependence of the results from the parameter $\rho$ is in excellent agreement with previous theoretical predictions.Comment: uuencoded latex file of 32 pages plus 3 ps figure

The critical behaviour of Ising spins on 2D Regge lattices

We performed a high statistics simulation of Ising spins coupled to 2D quantum gravity on toroidal geometries. The tori were triangulated using the Regge calculus approach and contained up to $512^2$ vertices. We used a constant area ensemble with an added $R^2$ interaction term, employing the $dl/l$ measure. We find clear evidence that the critical exponents of the Ising phase transition are consistent with the static critical exponents and do not depend on the coupling strength of the $R^2$ interaction term. We definitively can exclude for this type of model a behaviour as predicted by Boulatov and Kazakov [Phys. Lett. {\bf B186}, 379 (1987)] for Ising spins coupled to dynamically triangulated surfaces.Comment: 15 pages with 3 figures in form of an uudecoded compressed tar-ps-file. FUB-HEP 06/9

Dynamically Triangulated Ising Spins in Flat Space

A model describing Ising spins with short range interactions moving randomly in a plane is considered. In the presence of a hard core repulsion, which prevents the Ising spins from overlapping, the model is analogous to a dynamically triangulated Ising model with spins constrained to move on a flat surface. It is found that as a function of coupling strength and hard core repulsion the model exhibits multicritical behavior, with first and second order transition lines terminating at a tricritical point. The thermal and magnetic exponents computed at the tricritical point are consistent with the exact two-matrix model solution of the random Ising model, introduced previously to describe the effects of fluctuating geometries.Comment: (10 pages + 4 figures), CERN-Th-7577/9

Spiky Strings and Giant Holes

We analyse semiclassical strings in AdS in the limit of one large spin. In this limit, classical string dynamics is described by a finite number of collective coordinates corresponding to spikes or cusps of the string. The semiclassical spectrum consists of two branches of excitations corresponding to "large" and "small" spikes respectively. We propose that these states are dual to the excitations known as large and small holes in the spin chain description of N=4 SUSY Yang-Mills. The dynamics of large spikes in classical string theory can be mapped to that of a classical spin chain of fixed length. In turn, small spikes correspond to classical solitons propagating on the background formed by the large spikes. We derive the dispersion relation for these excitations directly in the finite gap formalism.Comment: 36 pages, 9 figure