The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation

We consider two-dimensional Bernoulli percolation at densityp>p _c and establish the following results:

The scaling window of the 2‐SAT transition

We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2‐clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n→α, the problem is known to have a phase transition at α_c=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite‐size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α_?(n,δ), α₊(n,δ)) such that the probability of satisfiability is greater than 1?δ for α<α_? and is less than δ for α>α₊. We show that W(n,δ)=(1?Θ(n^?1/3), 1+Θ(n^?1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n^1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(?Θ(nε³)) above the window, and goes to one like 1?Θ(n^?1|ε|^?3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2‐SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001     

Left and right convergence of graphs with bounded degree

The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left‐convergence), or counting homomorphisms into fixed graphs (right‐convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lovász, Sós and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case, if the set of graphs in the definition of right‐convergence is appropriately restricted. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left‐convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness. © 2012 Wiley Periodicals, Inc. Random Struct. 2012     

Large‐deviations/thermodynamic approach to percolation on the complete graph

We present a large‐deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large‐deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are “small.” One consequence is an immediate derivation of the “cavity” formula for the fraction of vertices in the giant component. As a byproduct of our analysis we compute the large‐deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only small components. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Partition Function Zeros at First-Order Phase Transitions: A General Analysis

We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [5]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Ising-like plus-minus symmetry, we also establish a local version of the Lee-Yang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves.Reproduction of the entire article for non-commercial purposes is permitted without charge.     

Partition Function Zeros at First-Order Phase Transitions: Pirogov—Sinai Theory

This paper is a continuation of our previous analysis(2) of partition functions zeros in models with first-order phase transitions and periodic boundary conditions. Here it is shown that the assumptions under which the results of ref. 2 were established are satisfied by a large class of lattice models. These models are characterized by two basic properties: The existence of only a finite number of ground states and the availability of an appropriate contour representation. This setting includes, for instance, the Ising, Potts, and Blume–Capel models at low temperatures. The combined results of ref. 2 and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class.     

A Proof of the Gibbs—Thomson Formula in the Droplet Formation Regime

We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs–Thomson formula for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general—albeit heuristic—reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.     

How to distribute antidote to control epidemics

We give a rigorous analysis of variations of the contact process on a finite graph in which the cure rate is allowed to vary from one vertex to the next, and even to depend on the current state of the system. In particular, we study the epidemic threshold in the models where the cure rate is proportional to the degree of the node or when it is proportional to the number of its infected neighbors. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Mean Field Analysis of Low–Dimensional Systems

For low–dimensional systems, (i.e. 2D and, to a certain extent, 1D) it is proved that mean–field theory can provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently “spead out” according to an exponential (Yukawa) potential it is shown that the energy, free energy and, in particular, the block magnetization (as defined on scales that are large compared with the lattice spacing but small compared to the range of the interaction) will only take on values near to those predicted by the associated mean–field theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the mean–field theory has a discontinuous phase transition featuring the breaking of a discrete symmetry then this sort of transition will occur in the actual system. Prominent examples include the two–dimensional q = 3 state Potts model. (b) If the mean–field theory has a discontinuous transition accompanied by the breaking of a continuous symmetry, the thermodynamic discontinuity is preserved even if the symmetry breaking is forbidden in the actual system. E.g. the two–dimensional O(3) nematic liquid crystal. Further it is demonstrated that mean–field behavior in the vicinity of the magnetic transition for layered Ising and XY systems also occurs in actual layered systems (with spread–out interactions) even if genuine magnetic ordering is precluded.     

Bethe lattice spin glass: The effects of a ferromagnetic bias and external fields. I. Bifurcation analysis

We present a rigorous analysis of the ±J Ising spin-glass model on the Bethe lattice with fixed uncorrelated boundary conditions. Phase diagrams are derived as a function of temperature vs. concentration of ferromagnetic bonds and, for a symmetric distribution of bonds, external field vs. temperature. In this part we characterize the bulk ordered phases using bifurcation theory: we prove the existence of a distribution of single-site magnetizations far inside the lattice which is stable with respect to changes in the boundary conditions.