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We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least $l^2$, for large $l$. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large $l$ this maximum is at least of order $l^{1/3}(\log l)^{-2/3}$. This complements the upper bound of order $l^{1/3}(\log l)^{2/3}$ known for the average local roughness. The exponent 1/3 on $l$ here is in keeping with predictions from the physics literature for interfaces in two dimensions., 28 pages, 1 figure (.eps file). See also http://math.usc.edu/~alexandr/

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model.

For lattice models on ℤ d , weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on ℤ2, including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on ℤ2, the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing.

A percolation process inRd is considered in which the sites are a Poisson process with intensity ρ and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other. The distribution of the number of sites in the clusterC of the origin is examined, and related to the geometry ofC. It is shown that when ρ andk are large, there is a characteristic radius λ such that conditionally on |C|=k, the convex hull ofC closely approximates a ball of radius λ, with high probability. When the normal volumek/ρ thatk points would occupy is small, the cluster is compressed, in that the number of points per unit volume in this λ-ball is much greater than the ambient density ρ. For larger normal volumes there is less compression. This can be compared to Bernoulli bond percolation on the square lattice in two dimensions, where an analog of this compression is known not to occur.

For two-dimensional Bernoulli percolation at densityp above the critical point, there exists a natural normg determined by the rate of decay of the connectivity function in every direction. IfW is the region of unit area with boundary of minimum possibleg-length, then it is known [4] that asN→∞, with probability approaching 1, conditionally onN≦|C(0)| 0, if γ is a curve enclosing a region of unit area such that the Hausdorff distanced H (γ+v,dW)≧δ for every translate γ+v, then theg-lengthg(γ) ≧g(∂W) +k δ2, at least for δ small.

Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array (φ nj ) of stochastic processes, indexed by a set S, to a sample-continuous Gaussian process, when the array satisfies a “random entropy” condition, analogous to one used by Gine and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-Ĉervonenkis property and each φ nj (f)fdνnj is of the form νnjc for some reasonable random finite signed measure v nj. As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik-Ĉervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.

Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-Cervonenkis class C of sets. These inequalities are particularly useful under the assumption P(∪{C∈C:P(C)