Holographic Algorithm with Matchgates Is Universal for Planar #\#CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Comment: 94 page

Equilibriumlike invaded cluster algorithm: critical exponents and dynamical properties

We present a detailed study of the Equilibriumlike invaded cluster algorithm (EIC), recently proposed as an extension of the invaded cluster (IC) algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is discussed in context of dynamical properties. We also discuss the relation to the Li-Sokal bound for the dynamical exponent $z$.Comment: 11 pages, 13 figures, accepted for publication in Phys. Rev.

Single-cluster dynamics for the random-cluster model

We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the $q$-state Potts model to non-integer values $q>1$. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer $q$, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents $z_{\rm exp} =0.07 (1), 0.521 (7)$, and $1.007 (9)$ for $q=2, 3$, and 4 respectively. For non-integer $q$, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.Comment: 7 figures, 4 table

Reformulation of the Stochastic Potential Switching Algorithm and a Generalized Fourtuin-Kasteleyn Representation

A new formulation of the stochastic potential switching algorithm is presented. This reformulation naturally leads us to a generalized Fourtuin-Kasteleyn representation of the partition function Z. A formula for internal energy E and that of heat capacity C are derived from derivatives of the partition function. We also derive a formula for the exchange probability in the replica exchange Monte Carlo method. By combining the formulae with the Stochastic Cutoff method, we can greatly reduce the computational time to perform internal energy and heat capacity measurements and the replica exchange Monte Carlo method in long-range interacting systems. Numerical simulations in three dimensional magnetic dipolar systems show the validity and efficiency of the method.Comment: 11 pages, 6 figures, to appear in PR

New critical frontiers for the Potts and percolation models

We obtain the critical threshold for a host of Potts and percolation models on lattices having a structure which permits a duality consideration. The consideration generalizes the recently obtained thresholds of Scullard and Ziff for bond and site percolation on the martini and related lattices to the Potts model and to other lattices.Comment: 9 pages, 5 figure

Critical frontier of the Potts and percolation models in triangular-type and kagome-type lattices I: Closed-form expressions

We consider the Potts model and the related bond, site, and mixed site-bond percolation problems on triangular-type and kagome-type lattices, and derive closed-form expressions for the critical frontier. For triangular-type lattices the critical frontier is known, usually derived from a duality consideration in conjunction with the assumption of a unique transition. Our analysis, however, is rigorous and based on an established result without the need of a uniqueness assumption, thus firmly establishing all derived results. For kagome-type lattices the exact critical frontier is not known. We derive a closed-form expression for the Potts critical frontier by making use of a homogeneity assumption. The closed-form expression is new, and we apply it to a host of problems including site, bond, and mixed site-bond percolation on various lattices. It yields exact thresholds for site percolation on kagome, martini, and other lattices, and is highly accurate numerically in other applications when compared to numerical determination.Comment: 22 pages, 13 figure

Directed Percolation and Generalized Friendly Walkers

We show that the problem of directed percolation on an arbitrary lattice is equivalent to the problem of m directed random walkers with rather general attractive interactions, when suitably continued to m=0. In 1+1 dimensions, this is dual to a model of interacting steps on a vicinal surface. A similar correspondence with interacting self-avoiding walks is constructed for isotropic percolation.Comment: 4 pages, 3 figures, to be published in Phys. Rev. Let

Crossing bonds in the random-cluster model

We derive the scaling dimension associated with crossing bonds in the random-cluster representation of the two-dimensional Potts model, by means of a mapping on the Coulomb gas. The scaling field associated with crossing bonds appears to be irrelevant, on the critical as well as on the tricritical branch. The latter result stands in a remarkable contrast with the existing result for the tricritical O(n) model that crossing bonds are relevant. In order to obtain independent confirmation of the Coulomb gas result for the crossing-bond exponent, we perform a finite-size-scaling analysis based on numerical transfer-matrix calculations.Comment: 2 figure

Short-time Dynamics of Percolation Observables

We consider the critical short-time evolution of magnetic and droplet-percolation order parameters for the Ising model in two and three dimensions, through Monte-Carlo simulations with the (local) heat-bath method. We find qualitatively different dynamic behaviors for the two types of order parameters. More precisely, we find that the percolation order parameter does not have a power-law behavior as encountered for the magnetization, but develops a scale (related to the relaxation time to equilibrium) in the Monte-Carlo time. We argue that this difference is due to the difficulty in forming large clusters at the early stages of the evolution. Our results show that, although the descriptions in terms of magnetic and percolation order parameters may be equivalent in the equilibrium regime, greater care must be taken to interprete percolation observables at short times. In particular, this concerns the attempts to describe the dynamics of the deconfinement phase transition in QCD using cluster observables.Comment: 5 pages, 4 figure

A generalized Kac-Ward formula

The Kac-Ward formula allows to compute the Ising partition function on a planar graph G with straight edges from the determinant of a matrix of size 2N, where N denotes the number of edges of G. In this paper, we extend this formula to any finite graph: the partition function can be written as an alternating sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of an orientable surface in which G embeds. We give two proofs of this generalized formula. The first one is purely combinatorial, while the second relies on the Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on geometric techniques. As a consequence of this second proof, we also obtain the following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the Ising model are one and the same.Comment: 23 pages, 8 figures; minor corrections in v2; to appear in J. Stat. Mech. Theory Ex