22 research outputs found
Random graph asymptotics on high-dimensional tori
We investigate the scaling of the largest critical percolation cluster on a
large d-dimensional torus, for nearest-neighbor percolation in high dimensions,
or when d>6 for sufficient spread-out percolation. We use a relatively simple
coupling argument to show that this largest critical cluster is, with high
probability, bounded above by a large constant times and below by a
small constant times , where V is the volume of the
torus. We also give a simple criterion in terms of the subcritical percolation
two-point function on Z^d under which the lower bound can be improved to small
constant times , i.e., we prove random graph asymptotics for the
largest critical cluster on the high-dimensional torus. This establishes a
conjecture by Aizenman (1997), apart from logarithmic corrections. We discuss
implications of these results on the dependence on boundary conditions for
high-dimensional percolation.
Our method is crucially based on the results by Borgs, Chayes, van der
Hofstad, Slade and Spencer (2005a, 2005b), where the scaling was
proved subject to the assumption that a suitably defined critical window
contains the percolation threshold on Z^d. We also strongly rely on mean-field
results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade
(1990) and Hara, van der Hofstad and Slade (2003).Comment: 22 page
The near-critical planar FK-Ising model
We study the near-critical FK-Ising model. First, a determination of the
correlation length defined via crossing probabilities is provided. Second, a
phenomenon about the near-critical behavior of FK-Ising is highlighted, which
is completely missing from the case of standard percolation: in any monotone
coupling of FK configurations (e.g., in the one introduced in
[Gri95]), as one raises near , the new edges arrive in a
self-organized way, so that the correlation length is not governed anymore by
the number of pivotal edges at criticality.Comment: 34 pages, 8 figures. This is a streamlined version; the previous one
contains more explanations and additional material on exceptional times in FK
models with general . Furthermore, the statement and proof of Theorem 1.2
have slightly change
One-dimensional infinite component vector spin glass with long-range interactions
We investigate zero and finite temperature properties of the one-dimensional
spin-glass model for vector spins in the limit of an infinite number m of spin
components where the interactions decay with a power, \sigma, of the distance.
A diluted version of this model is also studied, but found to deviate
significantly from the fully connected model. At zero temperature, defect
energies are determined from the difference in ground-state energies between
systems with periodic and antiperiodic boundary conditions to determine the
dependence of the defect-energy exponent \theta on \sigma. A good fit to this
dependence is \theta =3/4-\sigma. This implies that the upper critical value of
\sigma is 3/4, corresponding to the lower critical dimension in the
d-dimensional short-range version of the model. For finite temperatures the
large m saddle-point equations are solved self-consistently which gives access
to the correlation function, the order parameter and the spin-glass
susceptibility. Special attention is paid to the different forms of finite-size
scaling effects below and above the lower critical value, \sigma =5/8, which
corresponds to the upper critical dimension 8 of the hypercubic short-range
model.Comment: 27 pages, 27 figures, 4 table
On the size of the largest cluster in 2D critical percolation
We consider (near-)critical percolation on the square lattice. Let M_n be the
size of the largest open cluster contained in the box [-n,n]^2, and let pi(n)
be the probability that there is an open path from O to the boundary of the
box. It is well-known that for all 0< a < b the probability that M_n is smaller
than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are
bounded away from 0 as n tends to infinity. It is a natural question, which
arises for instance in the study of so-called frozen-percolation processes, if
a similar result holds for the probability that M_n is between an^2 pi(n) and
bn^2 pi(n). By a suitable partition of the box, and a careful construction
involving the building blocks, we show that the answer to this question is
affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the
argument.Comment: 12 pages, 3 figures, minor change
Upper bounds on the one-arm exponent for dependent percolation models
We prove upper bounds on the one-arm exponent for dependent
percolation models; while our main interest is level set percolation of smooth
Gaussian fields, the arguments apply to other models in the Bernoulli
percolation universality class, including Poisson-Voronoi and Poisson-Boolean
percolation. More precisely, in dimension we prove for
Gaussian fields with rapid correlation decay (e.g.\ the Bargmann-Fock field),
and in general dimensions we prove for finite-range fields and
for fields with rapid correlation decay. Although these
results are classical for Bernoulli percolation (indeed they are best-known in
general), existing proofs do not extend to dependent percolation models, and we
develop a new approach based on exploration and relative entropy arguments. We
also establish a new Russo-type inequality for smooth Gaussian fields which we
use to prove the sharpness of the phase transition for finite-range fields.Comment: 34 pages, 2 figure