Oriented Percolation in One-Dimensional 1/|x-y|^2 Percolation Models

We consider independent edge percolation models on Z, with edge occupation probabilities p_ = p if |x-y| = 1, 1 - exp{- beta / |x-y|^2} otherwise. We prove that oriented percolation occurs when beta > 1 provided p is chosen sufficiently close to 1, answering a question posed in [Commun. Math. Phys. 104, 547 (1986)]. The proof is based on multi-scale analysis.Comment: 19 pages, 2 figures. See also Commentary on J. Stat. Phys. 150, 804-805 (2013), DOI 10.1007/s10955-013-0702-

Dynamic Critical Behavior of the Chayes-Machta Algorithm for the Random-Cluster Model. I. Two Dimensions

We study, via Monte Carlo simulation, the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts ferromagnet to non-integer q \ge 1. We consider spatial dimension d=2 and 1.25 \le q \le 4 in steps of 0.25, on lattices up to 1024^2, and obtain estimates for the dynamic critical exponent z_{CM}. We present evidence that when 1 \le q \lesssim 1.95 the Ossola-Sokal conjecture z_{CM} \ge \beta/\nu is violated, though we also present plausible fits compatible with this conjecture. We show that the Li-Sokal bound z_{CM} \ge \alpha/\nu is close to being sharp over the entire range 1 \le q \le 4, but is probably non-sharp by a power. As a byproduct of our work, we also obtain evidence concerning the corrections to scaling in static observables.Comment: LaTeX2e, 75 pages including 26 Postscript figure

Stretched exponential relaxation for growing interfaces in quenched disordered media

We study the relaxation for growing interfaces in quenched disordered media. We use a directed percolation depinning model introduced by Tang and Leschhorn for 1+1-dimensions. We define the two-time autocorrelation function of the interface height C(t',t) and its Fourier transform. These functions depend on the difference of times t-t' for long enough times, this is the steady-state regime. We find a two-step relaxation decay in this regime. The long time tail can be fitted by a stretched exponential relaxation function. The relaxation time is proportional to the characteristic distance of the clusters of pinning cells in the direction parallel to the interface and it diverges as a power law. The two-step relaxation is lost at a given wave length of the Fourier transform, which is proportional to the characteristic distance of the clusters of pinning cells in the direction perpendicular to the interface. The stretched exponential relaxation is caused by the existence of clusters of pinning cells and it is a direct consequence of the quenched noise.Comment: 4 pages and 5 figures. Submitted (5/2002) to Phys. Rev.

Phase transitions with four-spin interactions

Using an extended Lee-Yang theorem and GKS correlation inequalities, we prove, for a class of ferromagnetic multi-spin interactions, that they will have a phase transition(and spontaneous magnetization) if, and only if, the external field $h=0$ (and the temperature is low enough). We also show the absence of phase transitions for some nonferromagnetic interactions. The FKG inequalities are shown to hold for a larger class of multi-spin interactions

Nonperturbative bound on high multiplicity cross sections in phi^4_3 from lattice simulation

We have looked for evidence of large cross sections at large multiplicities in weakly coupled scalar field theory in three dimensions. We use spectral function sum rules to derive bounds on total cross sections where the sum can be expresed in terms of a quantity which can be measured by Monte Carlo simulation in Euclidean space. We find that high multiplicity cross sections remain small for energies and multiplicities for which large effects had been suggested.Comment: 23 pages, revtex, seven eps figures revised version: typos corrected, some rewriting of discusion, same resul

Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex strip graphs $G$ of the honeycomb lattice for a variety of transverse widths equal to $L_y$ vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^m$, where $m$ denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for $N_{Z,G,j}$ for arbitrary $L_y$. We also present plots of zeros of the partition function in the $q$ plane for various values of $v$ and in the $v$ plane for various values of $q$. Explicit results for partition functions are given in the text for $L_y=2,3$ (free) and $L_y=4$ (cylindrical), and plots of partition function zeros are given for $L_y$ up to 5 (free) and $L_y=6$ (cylindrical). Plots of the internal energy and specific heat per site for infinite-length strips are also presented.Comment: 39 pages, 34 eps figures, 3 sty file

Phase coexistence of gradient Gibbs states

We consider the (scalar) gradient fields $\eta=(\eta_b)$--with $b$ denoting the nearest-neighbor edges in $\Z^2$--that are distributed according to the Gibbs measure proportional to \texte^{-\beta H(\eta)}\nu(\textd\eta). Here $H=\sum_bV(\eta_b)$ is the Hamiltonian, $V$ is a symmetric potential, $\beta>0$ is the inverse temperature, and $\nu$ is the Lebesgue measure on the linear space defined by imposing the loop condition $\eta_{b_1}+\eta_{b_2}=\eta_{b_3}+\eta_{b_4}$ for each plaquette $(b_1,b_2,b_3,b_4)$ in $\Z^2$. For convex $V$, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex $V$ undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature $\beta$. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., $E \eta_b=0$.Comment: 3 figs, PTRF style files include

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric

Study of Percolative Transitions with First-Order Characteristics in the Context of CMR Manganites

The unusual magneto-transport properties of manganites are widely believed to be caused by mixed-phase tendencies and concomitant percolative processes. However, dramatic deviations from "standard" percolation have been unveiled experimentally. Here, a semi-phenomenological description of Mn oxides is proposed based on coexisting clusters with smooth surfaces, as suggested by Monte Carlo simulations of realistic models for manganites, also briefly discussed here. The present approach produces fairly abrupt percolative transitions and even first-order discontinuities, in agreement with experiments. These transitions may describe the percolation that occurs after magnetic fields align the randomly oriented ferromagnetic clusters believed to exist above the Curie temperature in Mn oxides. In this respect, part of the manganite phenomenology could belong to a new class of percolative processes triggered by phase competition and correlations.Comment: 4 pages, 4 eps figure

Group testing with Random Pools: Phase Transitions and Optimal Strategy

The problem of Group Testing is to identify defective items out of a set of objects by means of pool queries of the form "Does the pool contain at least a defective?". The aim is of course to perform detection with the fewest possible queries, a problem which has relevant practical applications in different fields including molecular biology and computer science. Here we study GT in the probabilistic setting focusing on the regime of small defective probability and large number of objects, $p \to 0$ and $N \to \infty$. We construct and analyze one-stage algorithms for which we establish the occurrence of a non-detection/detection phase transition resulting in a sharp threshold, $\bar M$, for the number of tests. By optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling $\bar M\propto Np|\log p|$. Then we consider two-stages algorithms and analyze their performance for different choices of the first stage pools. In particular, via a proper random choice of the pools, we construct algorithms which attain the optimal value (previously determined in Ref. [16]) for the mean number of tests required for complete detection. We finally discuss the optimal pool design in the case of finite $p$