Gonihedric Ising Actions

We discuss a generalized Ising action containing nearest neighbour, next to nearest neighbour and plaquette terms that has been suggested as a potential string worldsheet discretization on cubic lattices by Savvidy and Wegner. This displays both first and second order transitions depending on the value of a ``self-intersection'' coupling as well as possessing a novel semi-global symmetry.Comment: Latex + 2 postscript figures. Poster session contribution to "Lattice96" conference, Washington University, StLoui

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

Softening Transitions with Quenched 2D Gravity

We perform extensive Monte Carlo simulations of the 10-state Potts model on quenched two-dimensional $\Phi^3$ gravity graphs to study the effect of quenched connectivity disorder on the phase transition, which is strongly first order on regular lattices. The numerical data provides strong evidence that, due to the quenched randomness, the discontinuous first-order phase transition of the pure model is softened to a continuous transition.Comment: 3 pages, LaTeX + 1 postscript figure. Talk presented at LATTICE96(other models). See also http://www.cond-mat.physik.uni-mainz.de/~janke/doc/home_janke.htm

Vertex Models on Feynman Diagrams

The statistical mechanics of spin models, such as the Ising or Potts models, on generic random graphs can be formulated economically by considering the N --> 1 limit of Hermitian matrix models. In this paper we consider the N --> 1 limit in complex matrix models, which describes vertex models of different sorts living on random graphs. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. We also make some remarks on vertex models on planar random graphs (the N --> infinity limit) where the resulting matrix models are not generally soluble using currently known methods. Nonetheless, some particular cases may be mapped onto known models and hence solved.Comment: 10 Pages text (LaTeX), 4 eps figure

Information Geometry and Phase Transitions

The introduction of a metric onto the space of parameters in models in Statistical Mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrization, the scalar curvature, R, plays a central role. A non-interacting model has a flat geometry (R=0), while R diverges at the critical point of an interacting one. Here, the information geometry is studied for a number of solvable statistical-mechanical models.Comment: 6 pages with 1 figur

The Phase Diagram of the Gonihedric 3d Ising Model via CVM

We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in ${\bf Z}^3$. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the self-avoidance coupling $\kappa$ is varied. The value of the magnetization critical exponent $\beta = 0.062 \pm 0.003$, calculated with the cluster variation--Pad\`e approximant method, is also close to the simulation results.Comment: 8 pages text (LaTex) + 3 eps figures bundled together with uufile

Spin Models on Thin Graphs

We discuss the utility of analytical and numerical investigation of spin models, in particular spin glasses, on ordinary ``thin'' random graphs (in effect Feynman diagrams) using methods borrowed from the ``fat'' graphs of two dimensional gravity. We highlight the similarity with Bethe lattice calculations and the advantages of the thin graph approach both analytically and numerically for investigating mean field results.Comment: Contribution to Parallel Session at Lattice95, 4 pages. Dodgy compressed ps file replaced with uuencoded LaTex original + ps figure

Geometrothermodynamics of black holes

The thermodynamics of black holes is reformulated within the context of the recently developed formalism of geometrothermodynamics. This reformulation is shown to be invariant with respect to Legendre transformations, and to allow several equivalent representations. Legendre invariance allows us to explain a series of contradictory results known in the literature from the use of Weinhold's and Ruppeiner's thermodynamic metrics for black holes. For the Reissner-Nordstr\"om black hole the geometry of the space of equilibrium states is curved, showing a non trivial thermodynamic interaction, and the curvature contains information about critical points and phase transitions. On the contrary, for the Kerr black hole the geometry is flat and does not explain its phase transition structure.Comment: Revised version, to be published in Gen.Rel.Grav.(Mashhoon's Festschrift

Balls in Boxes and Quantum Gravity

Four dimensional simplicial gravity has been studied by means of Monte Carlo simulations for some time, the main outcome of the studies being that the model undergoes a discontinuous phase transition between an elongated and a crumpled phase when one changes the curvature (Newton) coupling. In the crumpled phase there are singular vertices growing extensively with the volume of the system whereas the elongated phase resembles a branched-polymer. We have postulated that this behaviour is a manifestation of the constrained-mean-field scenario as realised in the Branched Polymer or Balls-in-Boxes model. These models share all the features of 4D simplicial gravity except that they exhibit a continuous phase transition. We note here that this defect can be remedied by a suitable choice of ensemble.Comment: 3 pages, LaTeX, 2 figures, uses espcrc2.sty, talk given at LATTICE9

Quenching 2D Quantum Gravity

We simulate the Ising model on a set of fixed random $\phi^3$ graphs, which corresponds to a {\it quenched} coupling to 2D gravity rather than the annealed coupling that is usually considered. We investigate the critical exponents in such a quenched ensemble and compare them with measurements on dynamical $\phi^3$ graphs, flat lattices and a single fixed $\phi^3$ graph.Comment: 8 page