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25
Conformally invariant scaling limits: an overview and collection of open problems
 Proceedings of the International Congress of Mathematicians, Madrid (M. SanzSolé et
, 2007
"... Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions beco ..."
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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a oneparameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
Every decision tree has an influential variable
 In Proceedings of the Fortysixth Annual Symposium on Foundations of Computer Science
, 2005
"... 0 ..."
TwoDimensional Scaling Limits via Marked Nonsimple Loops
, 2006
"... We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 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 ..."
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Cited by 12 (5 self)
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We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 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 nearcritical 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.
The scaling limit geometry of nearcritical 2D percolation
, 2006
"... We analyze the geometry of scaling limits of nearcritical 2D percolation, i.e., for p = pc + λδ 1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling l ..."
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Cited by 11 (2 self)
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We analyze the geometry of scaling limits of nearcritical 2D percolation, i.e., for p = pc + λδ 1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of “macroscopically pivotal ” lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a oneparameter family of nearcritical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.
Randomturn Hex and other selection games
 Amer. Math. Monthly
"... Overview. The game of Hex, invented independently by Piet Hein in 1942 and John Nash in 1948 [9], has two players who take turns placing stones of their respective colors on the hexagons of a rhombusshaped hexagonal grid (see Figure 1). A player wins by completing a path connecting the two opposite ..."
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Cited by 10 (5 self)
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Overview. The game of Hex, invented independently by Piet Hein in 1942 and John Nash in 1948 [9], has two players who take turns placing stones of their respective colors on the hexagons of a rhombusshaped hexagonal grid (see Figure 1). A player wins by completing a path connecting the two opposite sides of his or her color. Although it is easy to show
Dynamical stability of percolation for some interacting particle systems and fflmovability
, 2004
"... In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward εmovability which will be a key tool for our analysis. 1. Introduction. Consider ..."
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Cited by 10 (5 self)
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In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward εmovability which will be a key tool for our analysis. 1. Introduction. Consider
Uniqueness and nonuniqueness in percolation theory
, 2005
"... Abstract: This paper is an uptodate introduction to the problem of uniqueness versus nonuniqueness of infinite clusters for percolation on Z d and, more generally, on transitive graphs. For iid percolation on Z d, uniqueness of the infinite cluster is a classical result, while on certain other tr ..."
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Cited by 8 (1 self)
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Abstract: This paper is an uptodate introduction to the problem of uniqueness versus nonuniqueness of infinite clusters for percolation on Z d and, more generally, on transitive graphs. For iid percolation on Z d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models – most prominently the Fortuin–Kasteleyn randomcluster model – and in situations where the standard connectivity notion is replaced by entanglement or rigidity. Socalled simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
Dynamical models for circle covering: Brownian motion and
 Poisson updating, Annals of Prob. Volume 36, Number
"... We consider two dynamical variants of Dvoretzky’s classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of t ..."
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Cited by 7 (3 self)
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We consider two dynamical variants of Dvoretzky’s classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length ℓ is updated at rate ℓ −α where α≥0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c < 2 and there are times at which the circle is not fully covered if and only if c < 3. For the Poisson updating model, we obtain analogous results with c < α and c < α + 1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions. 1. Introduction.
On monochromatic arm exponents for 2d critical percolation, To appear
 B
, 1984
"... We investigate the socalled monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More s ..."
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Cited by 5 (1 self)
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We investigate the socalled monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic jarm exponent is strictly between the polychromatic jarm and (j +1)arm exponents. 1
The Fourier Spectrum of Critical Percolation
"... Consider the indicator function f of a twodimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exc ..."
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Cited by 5 (1 self)
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Consider the indicator function f of a twodimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the halfplane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.