Brownian Web and Oriented Percolation: Density Bounds

In a recent work, we proved that under diffusive scaling, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2 converges in distribution to the Brownian web. In that proof, the FKG inequality played an important role in establishing a density bound, which is a part of the convergence criterion for the Brownian web formulated by Fontes et al (2004). In this note, we illustrate how an alternative convergence criterion formulated by Newman et al (2005) can be verified in this case, which involves a dual density bound that can be established without using the FKG inequality. This alternative approach is in some sense more robust. We will also show that the spatial density of the collection of rightmost infinite open paths starting at time 0 decays asymptotically in time as c/\sqrt{t} for some c>0.Comment: 12 pages. This is a proceeding article for the RIMS workshop "Applications of Renormalization Group Methods in Mathematical Sciences", held at Kyoto University from September 12th to 14th, 2011. Submitted to the RIMS Kokyuroku serie

Two-Dimensional Scaling Limits via Marked Nonsimple Loops

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure

Diffusive scaling of the Kob-Andersen model in Zd\mathbb{Z}^d

We consider the Kob-Andersen model, a cooperative lattice gas with kinetic constraints which has been widely analyzed in the physics literature in connection with the study of the liquid/glass transition. We consider the model in a finite box of linear size $L$ with sources at the boundary. Our result, which holds in any dimension and significantly improves upon previous ones, establishes for any positive vacancy density $q$ a purely diffusive scaling of the relaxation time $T_{\rm rel}$ of the system. Furthermore, as $q\downarrow 0$ we prove upper and lower bounds on $L^{-2} T_{\rm rel} (q,L)$ which agree with the physicists belief that the dominant equilibration mechanism is a cooperative motion of rare large droplets of vacancies. The main tools combine a recent set of ideas and techniques developed to establish universality results for kinetically constrained spin models, with methods from bootstrap percolation, oriented percolation and canonical flows for Markov chains

Exact conjectured expressions for correlations in the dense O(1)(1) loop model on cylinders

We present conjectured exact expressions for two types of correlations in the dense O$(n=1)$ loop model on $L\times \infty$ square lattices with periodic boundary conditions. These are the probability that a point is surrounded by $m$ loops and the probability that $k$ consecutive points on a row are on the same or on different loops. The dense O$(n=1)$ loop model is equivalent to the bond percolation model at the critical point. The former probability can be interpreted in terms of the bond percolation problem as giving the probability that a vertex is on a cluster that is surrounded by \floor{m/2} clusters and \floor{(m+1)/2} dual clusters. The conjectured expression for this probability involves a binomial determinant that is known to give weighted enumerations of cyclically symmetric plane partitions and also of certain types of families of nonintersecting lattice paths. By applying Coulomb gas methods to the dense O$(n=1)$ loop model, we obtain new conjectures for the asymptotics of this binomial determinant.Comment: 17 pages, replaced by version accepted by JSTA

Magnetic strings as part of Yang-Mills plasma

Magnetic strings are defined as infinitely thin surfaces which are closed in the vacuum and can be open on an external monopole trajectory (that is, defined by 't Hooft loop). We review briefly lattice data on the magnetic strings which refer mostly to SU(2) and SU(3) pure Yang-Mills theories and concentrate on implications of the strings for the Yang-Mills plasma. We argue that magnetic strings might be a liquid component of the Yang-Mills plasma and suggest tests of this hypothesis.Comment: 15 pages, no figures, uses ws-procs9x6 style. Talk by V.I.Z. at SCGT06 workshop, Nagoya, Japan (November 2006

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036 without the appendice