The Poisson ratio of crystalline surfaces

A remarkable theoretical prediction for a crystalline (polymerized) surface is that its Poisson ratio (\sigma) is negative. Using a large scale Monte Carlo simulation of a simple model of such surfaces we show that this is indeed true. The precise numerical value we find is (\sigma \simeq -0.32) on a (128^2) lattice at bending rigidity (kappa = 1.1). This is in excellent agreement with the prediction (\sigma = -1/3) following from the self-consistent screening approximation of Le Doussal and Radzihovsky.Comment: 7 pages, 2 EPS figures, LaTeX2e. Revised version accepted for publication on Europhys. Let

Monte Carlo Renormalization of 2d Simplicial Quantum Gravity Coupled to Gaussian Matter

We extend a recently proposed real-space renormalization group scheme for dynamical triangulations to situations where the lattice is coupled to continuous scalar fields. Using Monte Carlo simulations in combination with a linear, stochastic blocking scheme for the scalar fields we are able to determine the leading eigenvalues of the stability matrix with good accuracy both for c = 1 and c = 10 theories.Comment: 17 pages, 7 figure

Unsupervised empirical Bayesian multiple testing with external covariates

In an empirical Bayesian setting, we provide a new multiple testing method, useful when an additional covariate is available, that influences the probability of each null hypothesis being true. We measure the posterior significance of each test conditionally on the covariate and the data, leading to greater power. Using covariate-based prior information in an unsupervised fashion, we produce a list of significant hypotheses which differs in length and order from the list obtained by methods not taking covariate-information into account. Covariate-modulated posterior probabilities of each null hypothesis are estimated using a fast approximate algorithm. The new method is applied to expression quantitative trait loci (eQTL) data.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS158 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Anisotropic Membranes

We describe the statistical behavior of anisotropic crystalline membranes. In particular we give the phase diagram and critical exponents for phantom membranes and discuss the generalization to self-avoiding membranes.Comment: LATTICE98(surfaces) 5 pages, 4 Postscript figure

Universality Classes of Self-Avoiding Fixed-Connectivity Membranes

We present an analysis of extensive large-scale Monte Carlo simulations of self-avoiding fixed-connectivity membranes for sizes (number of faces) ranging from 512 to 17672 (triangular) plaquettes. Self-avoidance is implemented via impenetrable plaquettes. We simulate the impenetrable plaquette model in both three and four bulk dimensions. In both cases we find the membrane to be flat for all temperatures: the size exponent in three dimensions is nu=0.95(5) (Hausdorff dimension d_H=2.1(1)). The single flat phase appears, furthermore, to be equivalent to the large bending rigidity phase of non-self-avoiding fixed-connectivity membranes - the roughness exponent in three dimensions is xi=0.63(4). This suggests that there is a unique universality class for flat fixed-connectivity membranes without attractive interactions. Finally we address some theoretical and experimental implications of our work

Minimal Dynamical Triangulations of Random Surfaces

We introduce and investigate numerically a minimal class of dynamical triangulations of two-dimensional gravity on the sphere in which only vertices of order five, six or seven are permitted. We show firstly that this restriction of the local coordination number, or equivalently intrinsic scalar curvature, leaves intact the fractal structure characteristic of generic dynamically triangulated random surfaces. Furthermore the Ising model coupled to minimal two-dimensional gravity still possesses a continuous phase transition. The critical exponents of this transition correspond to the usual KPZ exponents associated with coupling a central charge c=1/2 model to two-dimensional gravity.Comment: Latex, 9 pages, 3 figures, Published versio

New critical phenomena in 2d quantum gravity

We study $q=10$ and $q=200$ state Potts models on dynamical triangulated lattices and demonstrate that these models exhibit continuous phase transitions, contrary to the first order transition present on regular lattices. For $q=10$ the transition seems to be of 2nd order, while it seems to be of 3rd order for $q=200$. For $q=200$ the phase transition also induces a transition between typical fractal structures of the piecewise linear surfaces corresponding to the triangulations. The typical surface changes from having a tree-like structure to a fractal structure characterizing pure gravity when the temperature drops below the critical temperature. An investigation of the alignment of spin clusters shows that they are strongly correlated to the underlying fractal structure of the triangulated surfaces.Comment: 22 pages, uuencoded compressed ps-file. Use csh file.uu to get ps-fil

Numerical Observation of a Tubular Phase in Anisotropic Membranes

We provide the first numerical evidence for the existence of a tubular phase, predicted by Radzihovsky and Toner (RT), for anisotropic tethered membranes without self-avoidance. Incorporating anisotropy into the bending rigidity of a simple model of a tethered membrane with free boundary conditions, we show that the model indeed has two phase transitions corresponding to the flat-to-tubular and tubular-to-crumpled transitions. For the tubular phase we measure the Flory exponent $\nu_F$ and the roughness exponent $\zeta$. We find $\nu_F=0.305(14)$ and $\zeta=0.895(60)$, which are in reasonable agreement with the theoretical predictions of RT --- $\nu_F=1/4$ and $\zeta=1$.Comment: 8 pages, LaTeX, REVTEX, final published versio

A Universal Fractal Structure of 2D Quantum Gravity for c > 1

We investigate the fractal structure of $2d$ quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as $c \leq 1$. For $c >1$ it is not clear what distribution to expect. However, we observe strikingly similar distributions for various kinds of matter fields with the same $c$. This indicate that there might be some range of $c >1$ where the central charge of the matter fields alone determines the fractal structure of gravity coupled to matter. The hypothesis that the string susceptibility \g = 1/3 is found to be compatible with the data for $1 < c \leq 4$.Comment: 12 pages. compressed postscript file. Uncompressed size 3 Mb. Latex file without figures available by request. NBI-HE-93-6

Simulating Four-Dimensional Simplicial Gravity using Degenerate Triangulations

We extend a model of four-dimensional simplicial quantum gravity to include degenerate triangulations in addition to combinatorial triangulations traditionally used. Relaxing the constraint that every 4-simplex is uniquely defined by a set of five distinct vertexes, we allow triangulations containing multiply connected simplexes and distinct simplexes defined by the same set of vertexes. We demonstrate numerically that including degenerated triangulations substantially reduces the finite-size effects in the model. In particular, we provide a strong numerical evidence for an exponential bound on the entropic growth of the ensemble of degenerate triangulations, and show that a discontinuous crumpling transition is already observed on triangulations of volume N_4 ~= 4000.Comment: Latex, 8 pages, 4 eps-figure