46 research outputs found
Values of Brownian intersection exponents III: Two-sided exponents
This paper determines values of intersection exponents between packs of
planar Brownian motions in the half-plane and in the plane that were not
derived in our first two papers. For instance, it is proven that the exponent
describing the asymptotic decay of the probability of
non-intersection between two packs of three independent planar Brownian motions
each is . More generally, the values of and \tx (w_1', ..., w_k') are determined for all ,
, and all
. The proof relies on the results derived in our
first two papers and applies the same general methods. We first find the
two-sided exponents for the stochastic Loewner evolution processes in a
half-plane, from which the Brownian intersection exponents are determined via a
universality argument
One-arm exponent for critical 2D percolation
The probability that the cluster of the origin in critical site percolation
on the triangular grid has diameter larger than is proved to decay like
as
Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction
For random collections of self-avoiding loops in two-dimensional domains, we
define a simple and natural conformal restriction property that is
conjecturally satisfied by the scaling limits of interfaces in models from
statistical physics. This property is basically the combination of conformal
invariance and the locality of the interaction in the model. Unlike the Markov
property that Schramm used to characterize SLE curves (which involves
conditioning on partially generated interfaces up to arbitrary stopping times),
this property only involves conditioning on entire loops and thus appears at
first glance to be weaker.
Our first main result is that there exists exactly a one-dimensional family
of random loop collections with this property---one for each k in (8/3,4]---and
that the loops are forms of SLE(k). The proof proceeds in two steps. First,
uniqueness is established by showing that every such loop ensemble can be
generated by an "exploration" process based on SLE.
Second, existence is obtained using the two-dimensional Brownian loop-soup,
which is a Poissonian random collection of loops in a planar domain. When the
intensity parameter c of the loop-soup is less than 1, we show that the outer
boundaries of the loop clusters are disjoint simple loops (when c>1 there is
almost surely only one cluster) that satisfy the conformal restriction axioms.
We prove various results about loop-soups, cluster sizes, and the c=1 phase
transition.
Taken together, our results imply that the following families are equivalent:
1. The random loop ensembles traced by certain branching SLE(k) curves for k
in (8/3, 4].
2. The outer-cluster-boundary ensembles of Brownian loop-soups for c in (0,
1].
3. The (only) random loop ensembles satisfying the conformal restriction
axioms.Comment: This 91 page-long paper contains the previous versions (v2) of both
papers arxiv:1006.2373 and arxiv:1006.2374 that correspond to Part I and Part
II of the present paper. This merged longer paper is to appear in Annals of
Mathematic
Conformal fields, restriction properties, degenerate representations and SLE
In this note, we show how to relate the Schramm-Loewner Evolution processes
(SLE) to highest-weight representations of the Virasoro Algebra. The conformal
restriction properties of SLE that have been recently studied in the paper
arXiv:math.PR/0209343 by G. Lawler, O. Schramm and the second author play an
instrumental role. In this setup, various considerations from conformal field
theory can be interpreted and reformulated via SLE. This enables to make a
concrete link between the two-dimensional discrete critical systems from
statistical physics and conformal field theory.Comment: To appear in C.R.Acad. Sci. Paris, Ser. I Math. Minor modifications
from the first versio