Potts-Percolation-Gauss Model of a Solid

We study a statistical mechanics model of a solid. Neighboring atoms are connected by Hookian springs. If the energy is larger than a threshold the "spring" is more likely to fail, while if the energy is lower than the threshold the spring is more likely to be alive. The phase diagram and thermodynamic quantities, such as free energy, numbers of bonds and clusters, and their fluctuations, are determined using renormalization-group and Monte-Carlo techniques.Comment: 10 pages, 12 figure

Percolation on the average and spontaneous magnetization for q-states Potts model on graph

We prove that the q-states Potts model on graph is spontaneously magnetized at finite temperature if and only if the graph presents percolation on the average. Percolation on the average is a combinatorial problem defined by averaging over all the sites of the graph the probability of belonging to a cluster of a given size. In the paper we obtain an inequality between this average probability and the average magnetization, which is a typical extensive function describing the thermodynamic behaviour of the model

Poisson approximations for the Ising model

A $d$-dimensional Ising model on a lattice torus is considered. As the size $n$ of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field $a=a(n)$ tends to $-\infty$ and the pair potential $b$ remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case.Comment: 25 pages, 1 figur

Rejection-free Geometric Cluster Algorithm for Complex Fluids

We present a novel, generally applicable Monte Carlo algorithm for the simulation of fluid systems. Geometric transformations are used to identify clusters of particles in such a manner that every cluster move is accepted, irrespective of the nature of the pair interactions. The rejection-free and non-local nature of the algorithm make it particularly suitable for the efficient simulation of complex fluids with components of widely varying size, such as colloidal mixtures. Compared to conventional simulation algorithms, typical efficiency improvements amount to several orders of magnitude

Invaded cluster algorithm for equilibrium critical points

A new cluster algorithm based on invasion percolation is described. The algorithm samples the critical point of a spin system without a priori knowledge of the critical temperature and provides an efficient way to determine the critical temperature and other observables in the critical region. The method is illustrated for the two- and three-dimensional Ising models. The algorithm equilibrates spin configurations much faster than the closely related Swendsen-Wang algorithm.Comment: 13 pages RevTex and 4 Postscript figures. Submitted to Phys. Rev. Lett. Replacement corrects problem in printing figure

First-Order Phase Transition in Potts Models with finite-range interactions

We consider the $Q$-state Potts model on $\mathbb Z^d$, $Q\ge 3$, $d\ge 2$, with Kac ferromagnetic interactions and scaling parameter \ga. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for \ga small enough there is a value of the temperature at which coexist $Q+1$ Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for $d=2$, Q=3, in contrast with the case of nearest-neighbor interactions for which available results indicate a second order phase transition. Putting both results together provides an example of a system which undergoes a transition from second to first order phase transition by changing only the finite range of the interaction.Comment: Soumis pour publication a Journal of statistical physics - version r\'{e}vis\'{e}

Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0

Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex graph $G$ with $q$ colors, and considering the limiting function $W(\{G\},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, a fundamental question in graph theory is the following: is $W_r(\{G\},q) = q^{-1}W(\{G\},q)$ analytic or not at the origin of the $1/q$ plane? (where the complex generalization of $q$ is assumed). This question is also relevant in statistical mechanics because $W(\{G\},q)=\exp(S_0/k_B)$, where $S_0$ is the ground state entropy of the $q$-state Potts antiferromagnet on the lattice graph $\{G\}$, and the analyticity of $W_r(\{G\},q)$ at $1/q=0$ is necessary for the large-$q$ series expansions of $W_r(\{G\},q)$. Although $W_r$ is analytic at $1/q=0$ for many $\{G\}$, there are some $\{G\}$ for which it is not; for these, $W_r$ has no large-$q$ series expansion. It is important to understand the reason for this nonanalyticity. Here we give a general condition that determines whether or not a particular $W_r(\{G\},q)$ is analytic at $1/q=0$ and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with $W_r$ functions that are non-analytic at $1/q=0$ and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for $W_r(\{G\},q)$ to be analytic at $1/q=0$ is that $\{G\}$ is a regular lattice graph $\Lambda$. (This is known not to be a necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in Phys. Rev.

Absence of Phase Transition for Antiferromagnetic Potts Models via the Dobrushin Uniqueness Theorem

We prove that the $q$-state Potts antiferromagnet on a lattice of maximum coordination number $r$ exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever $q > 2r$. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay for $q \ge 7$), triangular lattice ($q \ge 11$), hexagonal lattice ($q \ge 4$), and Kagom\'e lattice ($q \ge 6$). The proofs are based on the Dobrushin uniqueness theorem.Comment: 32 pages including 3 figures. Self-unpacking file containing the tex file, the needed macros (epsf.sty, indent.sty, subeqnarray.sty, and eqsection.sty) and the 3 ps file

Theory of continuum percolation I. General formalism

The theoretical basis of continuum percolation has changed greatly since its beginning as little more than an analogy with lattice systems. Nevertheless, there is yet no comprehensive theory of this field. A basis for such a theory is provided here with the introduction of the Potts fluid, a system of interacting $s$-state spins which are free to move in the continuum. In the $s \to 1$ limit, the Potts magnetization, susceptibility and correlation functions are directly related to the percolation probability, the mean cluster size and the pair-connectedness, respectively. Through the Hamiltonian formulation of the Potts fluid, the standard methods of statistical mechanics can therefore be used in the continuum percolation problem.Comment: 26 pages, Late