No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1

Degree Distribution of Competition-Induced Preferential Attachment Graphs

We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cutoffs. This is the first explanation of how preferential attachment can arise from a more basic underlying mechanism of local competition. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a ``fertility'' parameter. We prove that the general model has a power-law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters. Limiting cases of the general model include the standard preferential attachment model without cutoff and the uniform attachment model.Comment: 24 pages, one figure. To appear in the journal: Combinatorics, Probability and Computing. Note, this is a long version, with complete proofs, of the paper "Competition-Induced Preferential Attachment" (cond-mat/0402268

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

Territorial Developments Based on Graffiti: a Statistical Mechanics Approach

We study the well-known sociological phenomenon of gang aggregation and territory formation through an interacting agent system defined on a lattice. We introduce a two-gang Hamiltonian model where agents have red or blue affiliation but are otherwise indistinguishable. In this model, all interactions are indirect and occur only via graffiti markings, on-site as well as on nearest neighbor locations. We also allow for gang proliferation and graffiti suppression. Within the context of this model, we show that gang clustering and territory formation may arise under specific parameter choices and that a phase transition may occur between well-mixed, possibly dilute configurations and well separated, clustered ones. Using methods from statistical mechanics, we study the phase transition between these two qualitatively different scenarios. In the mean-field rendition of this model, we identify parameter regimes where the transition is first or second order. In all cases, we have found that the transitions are a consequence solely of the gang to graffiti couplings, implying that direct gang to gang interactions are not strictly necessary for gang territory formation; in particular, graffiti may be the sole driving force behind gang clustering. We further discuss possible sociological -- as well as ecological -- ramifications of our results

Vertex Models and Random Labyrinths: Phase Diagrams for Ice-type Vertex Models

We propose a simple geometric recipe for constructing phase diagrams for a general class of vertex models obeying the ice rule. The disordered phase maps onto the intersecting loop model which is interesting in its own right and is related to several other statistical mechanical models. This mapping is also useful in understanding some ordered phases of these vertex models as they correspond to the polymer loop models with cross-links in their vulcanised phase.Comment: 8 pages, 6 figure

Revisiting the Theory of Finite Size Scaling in Disordered Systems: \nu Can Be Less Than 2/d

For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: \nu_{FS}<= 2/d. It is believed that the true critical exponent \nu of a disorder induced phase transition satisfies the same bound. We argue that in disordered systems the standard averaging introduces a noise, and a corresponding new diverging length scale, characterized by \nu_{FS}=2/d. This length scale, however, is independent of the system's own correlation length \xi. Therefore \nu can be less than 2/d. We illustrate these ideas on two exact examples, with \nu < 2/d. We propose a new method of disorder averaging, which achieves a remarkable noise reduction, and thus is able to capture the true exponents.Comment: 4 pages, Latex, one figure in .eps forma

Monte Carlo study of the Widom-Rowlinson fluid using cluster methods

The Widom-Rowlinson model of a fluid mixture is studied using a new cluster algorithm that is a generalization of the invaded cluster algorithm previously applied to Potts models. Our estimate of the critical exponents for the two-component fluid are consistent with the Ising universality class in two and three dimensions. We also present results for the three-component fluid.Comment: 13 pages RevTex and 2 Postscript figure

Invaded cluster simulations of the XY model in two and three dimensions

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000^2 and 120^3 respectively. A soft spin O(2) model, in the same universality class as the 3D XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K_c=0.45412(2) for the 3D XY model and eta=0.037(2) for the 3D XY universality class. For the 2D XY model the results are K_c=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the 2D or 3D XY models.Comment: 30 pages, 11 figures, problem viewing figures corrected in v

Dynamic and static properties of the invaded cluster algorithm

Simulations of the two-dimensional Ising and 3-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sub-lattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time tau(l,L) for an l*l sub-lattice attains a maximum in the crossover region, and a dynamic exponent z for the IC algorithm is defined according to tau_max ~ L^z. Simulation results for the 3-state model yield z=.346(.002) which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that z<.21 and possibly that tau_max ~ log L so that z=0 -- similar to previous results for the SW and Wolff algorithms.Comment: 21 pages with 12 figure