2,855 research outputs found
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
Detection of an anomalous cluster in a network
We consider the problem of detecting whether or not, in a given sensor
network, there is a cluster of sensors which exhibit an "unusual behavior."
Formally, suppose we are given a set of nodes and attach a random variable to
each node. We observe a realization of this process and want to decide between
the following two hypotheses: under the null, the variables are i.i.d. standard
normal; under the alternative, there is a cluster of variables that are i.i.d.
normal with positive mean and unit variance, while the rest are i.i.d. standard
normal. We also address surveillance settings where each sensor in the network
collects information over time. The resulting model is similar, now with a time
series attached to each node. We again observe the process over time and want
to decide between the null, where all the variables are i.i.d. standard normal,
and the alternative, where there is an emerging cluster of i.i.d. normal
variables with positive mean and unit variance. The growth models used to
represent the emerging cluster are quite general and, in particular, include
cellular automata used in modeling epidemics. In both settings, we consider
classes of clusters that are quite general, for which we obtain a lower bound
on their respective minimax detection rate and show that some form of scan
statistic, by far the most popular method in practice, achieves that same rate
to within a logarithmic factor. Our results are not limited to the normal
location model, but generalize to any one-parameter exponential family when the
anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions
In this paper we prove results on Birkhoff and Oseledets genericity along
certain curves in the space of affine lattices and in moduli spaces of
translation surfaces. We also prove applications of these results to dynamical
billiards, mathematical physics and number theory. In the space of affine
lattices , we prove that almost every
point on a curve with some non-degeneracy assumptions is Birkhoff generic for
the geodesic flow. This implies almost everywhere genericity for some curves in
the locus of branched covers of the torus inside the stratum
of translation surfaces. For these curves (and more in general curves which are
well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff
genericity) we also prove that almost every point is Oseledets generic for the
Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin.
As applications, we first consider a class of pseudo-integrable billiards,
billiards in ellipses with barriers, which was recently explored by Dragovic
and Radnovic, and prove that for almost every parameter, the billiard flow is
uniquely ergodic within the region of phase space in which it is trapped. We
then consider any periodic array of Eaton retroreflector lenses, placed on
vertices of a lattice, and prove that in almost every direction light rays are
each confined to a band of finite width. This generalizes a phenomenon recently
discovered by Fraczek and Schmoll which could so far only be proved for random
periodic configurations. Finally, a result on the gap distribution of
fractional parts of the sequence of square roots of positive integers, which
extends previous work by Elkies and McMullen, is also obtained.Comment: To appear in Journal of Modern Dynamic
Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE
This is the first part of a work aimed at constructing the stress-energy
tensor of conformal field theory as a local "object" in conformal loop
ensembles (CLE). This work lies in the wider context of re-constructing quantum
field theory from mathematically well-defined ensembles of random objects. The
goal of the present paper is two-fold. First, we provide an introduction to
CLE, a mathematical theory for random loops in simply connected domains with
properties of conformal invariance, developed recently by Sheffield and Werner.
It is expected to be related to CFT models with central charges between 0 and 1
(including all minimal models). Second, we further develop the theory by
deriving results that will be crucial for the construction of the stress-energy
tensor. We introduce the notions of support and continuity of CLE events, about
which we prove basic but important theorems. We then propose natural
definitions of CLE probability functions on the Riemann sphere and on doubly
connected domains. Under some natural assumptions, we prove conformal
invariance and other non-trivial theorems related to these constructions. We
only use the defining properties of CLE as well as some basic results about the
CLE measure. Although this paper is guided by the construction of the
stress-energy tensor, we believe that the theorems proved and techniques used
are of interest in the wider context of CLE. The actual construction will be
presented in the second part of this work.Comment: 61 pages, 10 figures; v2: typos/math corrected, steps in proofs
added, one theorem added, one theorem strenghtene
On the scaling limits of planar percolation
We prove Tsirelson's conjecture that any scaling limit of the critical planar
percolation is a black noise. Our theorems apply to a number of percolation
models, including site percolation on the triangular grid and any subsequential
scaling limit of bond percolation on the square grid. We also suggest a natural
construction for the scaling limit of planar percolation, and more generally of
any discrete planar model describing connectivity properties.Comment: With an Appendix by Christophe Garban. Published in at
http://dx.doi.org/10.1214/11-AOP659 the Annals of Probability
(http://www.imstat.org/aop/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Random curves, scaling limits and Loewner evolutions
61 pages, 26 figuresIn this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally- invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves; moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.Peer reviewe
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