609 research outputs found
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
On the Kert\'esz line: Some rigorous bounds
We study the Kert\'esz line of the --state Potts model at (inverse)
temperature , in presence of an external magnetic field . This line
separates two regions of the phase diagram according to the existence or not of
an infinite cluster in the Fortuin-Kasteleyn representation of the model. It is
known that the Kert\'esz line coincides with the line of first
order phase transition for small fields when is large enough. Here we prove
that the first order phase transition implies a jump in the density of the
infinite cluster, hence the Kert\'esz line remains below the line of first
order phase transition. We also analyze the region of large fields and prove,
using techniques of stochastic comparisons, that equals to the leading order, as goes to
where is the threshold for bond percolation.Comment: 11 pages, 1 figur
Graphical Representations for Ising Systems in External Fields
A graphical representation based on duplication is developed that is suitable
for the study of Ising systems in external fields. Two independent replicas of
the Ising system in the same field are treated as a single four-state
(Ashkin-Teller) model. Bonds in the graphical representation connect the
Ashkin-Teller spins. For ferromagnetic systems it is proved that ordering is
characterized by percolation in this representation. The representation leads
immediately to cluster algorithms; some applications along these lines are
discussed.Comment: 13 pages amste
Lebowitz Inequalities for Ashkin-Teller Systems
We consider the Ashkin-Teller model with negative four-spin coupling but
still in the region where the ground state is ferromagnetic. We establish the
standard Lebowitz inequality as well as the extension that is necessary to
prove a divergent susceptibility.Comment: Ams-TeX, 12 pages; two references added, final version accepted for
publication in Physica
Mean Field Analysis of LowâDimensional Systems
For lowâdimensional systems, (i.e. 2D and, to a certain extent, 1D) it is proved that meanâfield theory can provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently âspead out â 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 meanâfield theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the meanâfield 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 twoâdimensional q = 3 state Potts model. (b) If the meanâfield 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 twoâdimensional O(3) nematic liquid crystal. Further it is demonstrated that meanâfield behavior in the vicinity of the magnetic transition for layered Ising and XY systems also occurs in actual layered systems (with spreadâout interactions) even if genuine magnetic ordering is precluded
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