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

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

Cluster Monte Carlo study of multi-component fluids of the Stillinger-Helfand and Widom-Rowlinson type

Phase transitions of fluid mixtures of the type introduced by Stillinger and Helfand are studied using a continuum version of the invaded cluster algorithm. Particles of the same species do not interact, but particles of different types interact with each other via a repulsive potential. Examples of interactions include the Gaussian molecule potential and a repulsive step potential. Accurate values of the critical density, fugacity and magnetic exponent are found in two and three dimensions for the two-species model. The effect of varying the number of species and of introducing quenched impurities is also investigated. In all the cases studied, mixtures of $q$-species are found to have properties similar to $q$-state Potts models.Comment: 25 pages, 5 figure

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

Rejoinder to the Response arXiv:0812.2330 to 'Comment on a recent conjectured solution of the three-dimensional Ising model'

We comment on Z. D. Zhang's Response [arXiv:0812.2330] to our recent Comment [arXiv:0811.3876] addressing the conjectured solution of the three-dimensional Ising model reported in [arXiv:0705.1045].Comment: 2 page

Avoided Critical Behavior in O(n) Systems

Long-range frustrating interactions, even if their strength is infinitesimal, can give rise to a dramatic proliferations of ground or near-ground states. As a consequence, the ordering temperature can exhibit a discontinuous drop as a function of the frustration. A simple model of the doped Mott insulator, where the short-range tendency of the holes to phase separate competes with long-range Coulomb effects, exhibits this "avoided critical" behavior. This model may serve as a paradigm for many other systems.Comment: 4 pages, 2 figure

Density of some titanium-bearing silicate liquids and the compositional dependence of the partial molar volume of TiO2

The densities of thirteen silicate liquids along the Na2SiO3-TiO2 and CaSiO3-TiO2 joins and six other titanium-bearing silicate liquids of the general formula TiSiO5 (where X = Li, Na, K, Rb, Cs, Ca, Sr, Ba) have been measured in equilibrium with air using the double Pt bob Archimedean method. The Na2SiO3-TiO2 join was investigated from 10–50 mole% TiO2 in the temperature range 1000–1150°C whereas the CaSiO3-TiO2 join was investigated from 10–80 mole% TiO2 in the temperature range of 1400–1625°C. Density increases with TiO2 content along both joins. Partial molar volumes of the binary endmembers, Na2SiO3 and CaSiO3, and of TiO2 have been computed. The partial molar volume of Na2SiO3 agrees well with that determined by Bockris et al. (1955). The partial molar volume of CaSiO3 is in disagreement with that of Tomlinson et al. (1958). The partial molar volume of TiO2 derived from a linear fit to the Na2SiO3-TiO2 join is 27.6(3) cm3/mole at 1150°C. The partial molar volume of TiO2 derived from linear extrapolation of the CaSiO3-TiO2 data to TiO2 at 1600°C is 24.3(4) cm3/mole. Comparison of the partial molar volume data from these binary joins with TiO2 liquid density data (Dingwell, 1991) requires the existence of a large positive excess volume in the Na2SiO3-TiO2 system at 1150°C