Scaling Limit for the Incipient Spanning Clusters
نویسنده
چکیده
Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the direct description of the limiting continuum theory. The resulting structure is expected to exhibit strict conformal invariance, and facilitate the mathematical discussion of questions related to universality of critical behavior, conformal invariance, and some relations with a number of field theories.
منابع مشابه
On the Number of Incipient Spanning Clusters
In critical percolation models, in a large cube there will typically be more than one cluster of comparable diameter. In 2D, the probability of k >> 1 spanning clusters is of the order e k 2 . In dimensions d > 6, when η = 0 the spanning clusters proliferate: for L → ∞ the spanning probability tends to one, and there typically are ≈ L spanning clusters of size comparable to |Cmax| ≈ L. The rigo...
متن کاملThe Number of Incipient Spanning Clusters in Two-Dimensional Percolation
Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large. The study of the structure of large clusters at the percolation threshold continues to pose interesting problems whose solution sheds light on the nature of the critical state in general. Recent...
متن کاملInfinite canonical super-Brownian motion and scaling limits
We construct a measure valued Markov process which we call infinite canonical superBrownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on Zd when these objects are (a) critical; (b) mean-field and (c) infini...
متن کاملProbability of Incipient Spanning Clusters in Critical Two-Dimensional Percolation
It was a common belief until a very recent time that on two-dimensional (2D) lattices at percolation threshold pc there exists exactly one percolation cluster [1,2]. New insight developed recently by Aizenman, who proved [3] that the number of Incipient Spanning Clusters (ISC) in 2D critical percolation can be larger than one, and that the probability of at least k separate clusters is bounded ...
متن کاملUniversal crossing probabilities and incipient spanning clusters in directed percolation
Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the sys...
متن کامل