59 research outputs found
Corner percolation on and the square root of 17
We consider a four-vertex model introduced by B\'{a}lint T\'{o}th: a
dependent bond percolation model on in which every edge is
present with probability 1/2 and each vertex has exactly two incident edges,
perpendicular to each other. We prove that all components are finite cycles
almost surely, but the expected diameter of the cycle containing the origin is
infinite. Moreover, we derive the following critical exponents: the tail
probability (diameter of the cycle of the origin )
and the expectation (length of a typical cycle with
diameter , with and
The value of comes from a singular
sixth order ODE, while the relation corresponds to the fact
that the scaling limit of the natural height function in the model is the
additive Brownian motion, whose level sets have Hausdorff dimension 3/2. We
also include many open problems, for example, on the conformal invariance of
certain linear entropy models.Comment: Published in at http://dx.doi.org/10.1214/07-AOP373 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Noise sensitivity in bootstrap percolation
Answering questions of Itai Benjamini, we show that the event of complete
occupation in 2-neighbour bootstrap percolation on the d-dimensional box [n]^d,
for d\geq 2, at its critical initial density p_c(n), is noise sensitive, while
in k-neighbour bootstrap percolation on the d-regular random graph G_{n,d}, for
2\leq k\leq d-2, it is insensitive. Many open problems remain.Comment: 16 page
Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster
In dynamical critical site percolation on the triangular lattice or bond
percolation on \Z^2, we define and study a local time measure on the
exceptional times at which the origin is in an infinite cluster. We show that
at a typical time with respect to this measure, the percolation configuration
has the law of Kesten's Incipient Infinite Cluster. In the most technical
result of this paper, we show that, on the other hand, at the first exceptional
time, the law of the configuration is different. We also study the collapse of
the infinite cluster near typical exceptional times, and establish a relation
between static and dynamic exponents, analogous to Kesten's near-critical
relation.Comment: 56 pages, 7 figures. In this version, we prove convergence only for
one of our local time constructions. Consequently, some arguments are
slightly changed in Sections 2 and
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio
The scaling limits of near-critical and dynamical percolation
We prove that near-critical percolation and dynamical percolation on the
triangular lattice have a scaling limit as the mesh , in the "quad-crossing" space of percolation configurations
introduced by Schramm and Smirnov. The proof essentially proceeds by
"perturbing" the scaling limit of the critical model, using the pivotal
measures studied in our earlier paper. Markovianity and conformal covariance of
these new limiting objects are also established.Comment: 72 pages, 7 figures. Slightly revised, final versio
- …