Browsing by Subject "CRITICAL PERCOLATION"

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  • Flores, S. M.; Simmons, J. J. H.; Kleban, P.; Ziff, R. M. (2017)
    In this article, we use our results from Flores and Kleban (2015 Commun. Math. Phys. 333 389-434, 2015 Commun. Math. Phys. 333 435-81, 2015 Commun. Math. Phys. 333 597-667, 2015 Commun. Math. Phys. 333 669715) to generalize known formulas for crossing probabilities. Prior crossing results date back to Cardy's prediction of a formula for the probability that a percolation cluster in two dimensions connects the left and right sides of a rectangle at the percolation critical point in the continuum limit (Cardy 1992 J. Phys. A: Math. Gen. 25 L201-6). Here, we predict a new formula for crossing probabilities of a continuum limit loop-gas model on a planar lattice inside a 2N-sided polygon. In this model, boundary loops exit and then re-enter the polygon through its vertices, with exactly one loop passing once through each vertex, and these loops join the vertices pairwise in some specified connectivity through the polygon's exterior. The boundary loops also connect the vertices through the interior, which we regard as a crossing event. For particular values of the loop fugacity, this formula specializes to FK cluster (resp. spin cluster) crossing probabilities of a critical Q-state random cluster (resp. Potts) model on a lattice inside the polygon in the continuum limit. This includes critical percolation as the Q = 1 random cluster model. These latter crossing probabilities are conditioned on a particular side-alternating free/fixed (resp. fluctuating/fixed) boundary condition on the polygon's perimeter, related to how the boundary loops join the polygon's vertices pairwise through the polygon's exterior in the associated loop-gas model. For Q is an element of{2, 3, 4}, we compare our predictions of these random cluster (resp. Potts) model crossing probabilities in a rectangle (N = 2) and in a hexagon (N = 3) with high-precision computer simulation measurements. We find that the measurements agree with our predictions very well for Q is an element of{2, 3} and reasonably well if Q = 4.
  • Kemppainen, Antti; Smirnov, Stanislav (2019)
    In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces. Some other results in this article are convergence of the exploration process of the loop ensemble (or the branch of the exploration tree) to SLE(κ,κ−6), κ=16/3, and convergence of a generalization of this process for 4 marked points to SLE[κ,Z], κ=16/3, where Z refers to a partition function. The latter SLE process is a process that can't be written as a SLE(κ,ρ1,ρ2,…) process, which are the most commonly considered generalizations of SLEs.
  • Kemppainen, Antti; Smirnov, Stanislav (2017)
    In 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.