Schramm’s proof of Watts’ formula

@article{Sheffield2011SchrammsPO,
  title={Schramm’s proof of Watts’ formula},
  author={Scott Sheffield and David Bruce Wilson},
  journal={Annals of Probability},
  year={2011},
  volume={39},
  pages={1844-1863}
}
G\'{e}rard Watts predicted a formula for the probability in percolation that there is both a left--right and an up--down crossing, which was later proved by Julien Dub\'{e}dat. Here we present a simpler proof due to Oded Schramm, which builds on Cardy's formula in a conceptually appealing way: the triple derivative of Cardy's formula is the sum of two multi-arm densities. The relative sizes of the two terms are computed with Girsanov conditioning. The triple integral of one of the terms is… 
View PDF on arXiv
18 Citations
Highly Influential Citations
1
Background Citations
7

Figures from this paper

18 Citations

Citation Type

Citation Type

Oded Schramm: From Circle Packing to SLE
When I first met Oded Schramm in January 1991 at the University of California, San Diego, he introduced himself as a “Circle Packer”. This modest description referred to his Ph.D. thesis around the…
A formula for crossing probabilities of critical systems inside polygons
In this article, we generalize known formulas for crossing probabilities. Prior crossing results date back to J. Cardy's prediction of a formula for the probability that a percolation cluster in two…
Boundary Correlations in Planar LERW and UST
We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show…
A Proof of Factorization Formula for Critical Percolation
We give mathematical proofs to a number of statements which appeared in the series of papers by Simmons et al. (Phys Rev E 76(4):041106, 2007; J Stat Mech Theory Exp 2009(2):P02067, 33, 2009) where…
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal…
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal…
The expected number of critical percolation clusters intersecting a line segment
We study critical percolation on a regular planar lattice. Let EG(n) be the expected number of open clusters intersecting or hitting the line segment [0; n]. (For the subscript G we either take H,…
Factorization of correlations in two-dimensional percolation on the plane and torus
Recently, Delfino and Viti have examined the factorization of the threepoint density correlation function P3 at the percolation point in terms of the two-point density correlation functions P2.…
Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotics properties
Applying the quantum group method developed in [KP20], we construct solutions to the Benoit & Saint-Aubin partial differential equations with boundary conditions given by specific recursive…
Pivotal, cluster, and interface measures for critical planar percolation
This work is the first in a series of papers devoted to the construction and study of scaling limits of dynamical and near-critical planar percolation and related objects like invasion percolation…
...
1
2
...

References

SHOWING 1-10 OF 18 REFERENCES
On Crossing Event Formulas in Critical Two-Dimensional Percolation
Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts's formula for the horizontal-vertical crossing…
Excursion decompositions for SLE and Watts' crossing formula
It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if κ>4 and a.s. cutpoints if 4<κ<8. If κ>4, an appropriate version of SLE(κ) has a renewal property: it starts afresh…
Critical percolation exploration path and SLE6: 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 SLE6. We provide here a detailed…
Euler Integrals for Commuting SLEs
Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability…
Multiple Schramm–Loewner Evolutions and Statistical Mechanics Martingales
A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at…
Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
Conformal invariance of planar loop-erased random walks and uniform spanning trees
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\mathop \subset \limits_ \ne \mathbb{C} \) is equal to the radial SLE2 path. In particular, the…
LETTER TO THE EDITOR: A crossing probability for critical percolation in two dimensions
Langlands et al considered two crossing probabilities, and , in their extensive numerical investigations of critical percolation in two dimensions. Cardy was able to find the exact form of by…
Reversibility of chordal SLE
We prove that the chordal SLEκ trace is reversible for κ ∈ (0, 4]. 1. Introduction. Stochastic Loewner evolutions (SLEs) are introduced by Oded Schramm [11] to describe the scaling limits of some…
On the universality of crossing probabilities in two-dimensional percolation
Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of widtha and heightb are superimposed on the…
...
1
2
...