67 research outputs found
Towards Conformal Invariance and a Geometric Representation of the 2D Ising Magnetization Field
We study the continuum scaling limit of the critical Ising magnetization in
two dimensions. We prove the existence of subsequential limits, discuss
connections with the scaling limit of critical FK clusters, and describe work
in progress of the author with C. Garban and C.M. Newman.Comment: 20 pages, 1 figure, presented at the workshop "Inhomogeneous Random
Systems" held at IHP (Paris) on January 26-27, 201
Universal Behavior of Connectivity Properties in Fractal Percolation Models
Partially motivated by the desire to better understand the connectivity phase
transition in fractal percolation, we introduce and study a class of continuum
fractal percolation models in dimension d greater than or equal to 2. These
include a scale invariant version of the classical (Poisson) Boolean model of
stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler
and Werner. The models lead to random fractal sets whose connectivity
properties depend on a parameter lambda. In this paper we mainly study the
transition between a phase where the random fractal sets are totally
disconnected and a phase where they contain connected components larger than
one point. In particular, we show that there are connected components larger
than one point at the unique value of lambda that separates the two phases
(called the critical point). We prove that such a behavior occurs also in
Mandelbrot's fractal percolation in all dimensions d greater than or equal to
2. Our results show that it is a generic feature, independent of the dimension
or the precise definition of the model, and is essentially a consequence of
scale invariance alone. Furthermore, for d=2 we prove that the presence of
connected components larger than one point implies the presence of a unique,
unbounded, connected component.Comment: 29 pages, 4 figure
Critical Percolation Exploration Path and SLE(6): a Proof of Convergence
It was argued by Schramm and Smirnov that the critical site percolation
exploration path on the triangular lattice converges in distribution to the
trace of chordal SLE(6). We provide here a detailed proof, which relies on
Smirnov's theorem that crossing probabilities have a conformally invariant
scaling limit (given by Cardy's formula). The version of convergence to SLE(6)
that we prove suffices for the Smirnov-Werner derivation of certain critical
percolation crossing exponents and for our analysis of the critical percolation
full scaling limit as a process of continuum nonsimple loops.Comment: 45 pages, 14 figures; revised version following the comments of a
refere
Continuum Nonsimple Loops and 2D Critical Percolation
Substantial progress has been made in recent years on the 2D critical
percolation scaling limit and its conformal invariance properties. In
particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter k=6)
was, in the work of Schramm and of Smirnov, identified as the scaling limit of
the critical percolation ``exploration process.'' In this paper we use that and
other results to construct what we argue is the full scaling limit of the
collection of all closed contours surrounding the critical percolation clusters
on the 2D triangular lattice. This random process or gas of continuum nonsimple
loops in the plane is constructed inductively by repeated use of chordal SLE6.
These loops do not cross but do touch each other -- indeed, any two loops are
connected by a finite ``path'' of touching loops.Comment: 16 pages, 3 figure
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