A Percolation Formula

By Oded Schramm. 2001

171 citations recorded by the source corpus.

Let $A$ be an arc on the boundary of the unit disk $U$. We prove an asymptotic formula for the probability that there is a percolation cluster $K$ for critical site percolation on the triangular grid in $U$ which intersects $A$ and such that $0$ is surrounded by the union of $K$ and $A$.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9691956d17e1ad7203d4b5f506176d0d9489323c

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Citations in the corpus

Harmonic explorer and its convergence to SLE4 (2003, background): “tends to either zero or 1 as t tends to infinity, depending on whether z is on the left or the right side of the path (see [15], Lemma 3).”
Commutation relations for Schramm‐Loewner evolutions (2004, background): “right); this can be made explicit (see [20]).”
Conformal restriction: The chordal case (2002, background): “Similarly, the determination of the critical exponents for SLE6 in [23, 24, 25] combined with Smirnov’s [46] proof of conformal invariance for critical percolation on the triangular lattice (along with Kesten’s hyperscaling relations) facilitated proofs of several fundamental properties of critical ”
Towards conformal invariance of 2D lattice models (2007, background): “In that case the probability of SLE(4) passing to one side of a point z is harmonic in z and has boundary conditions 0 and 1, see [33].”
2D growth processes: SLE and Loewner chains (2006, methodology): “ion in an annulus have also been predicted by Cardy using CFT [30]. Their SLE companions have been derived in [44]. Finally, another excursion formula for percolation has been proved 119 using SLE in [118]. The one-arm exponent which governs the decrease of the probability that the critical percolat”
Conformal invariance of planar loop-erased random walks and uniform spanning trees (2001, background): “This was the case for critical site percolation on the triangular grid [Smi01a, Sch01 , LSW02a, SW01].”
Schramm-Loewner Evolution (2007, unknown): “[71] O. Schramm (2001), A percolation formula, Electron.”
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, unknown): “By Lemma 3 in [24], we see that λ ∈ D(z) if and only if z / ∈ H+(γ).”
Conformal invariance in two-dimensional turbulence (2006, unknown): “pdf [27] Schramm, O.”
CRITICAL EXPONENTS FOR TWO-DIMENSIONAL PERCOLATION (2001, background): “Remark 2 (Changing the colors) The convention to study blue crossings is not restrictive: it is standard (see e.g., [2, 15]) that for any given sequence of colours, the probability that there exist left-right crossings of a topological rectangle (for discrete critical percolation) of some…”
Conformal Field Theories of Stochastic Loewner Evolutions (2002, methodology): “l ﬁelds. Since we use operator product expansion properties [2] of conformal ﬁelds to show that the vectors hvE| satisfy the appropriate requirements, our apporach is more algebraic than that of refs.[14, 18, 19] but less rigorous. 19 5.1 Generalized Cardy’s formula. The most famous crossing probabi”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “the probability measure Qcan be constructed as follows: First sample b0(γ); this deﬁnes its tip z0. Then draw an SLEκ (for some given value of κ) from z0 to 0− in H\b0. Standard SLE computations (see [36]) show that the probability that an SLEκfrom 1 to 0 in Hhits the (semi-)circle 54 of radius ρ&gt”
Random planar curves and Schramm-Loewner evolutions (2003, methodology): “the convergence of the discrete exploration process to SLE6 (announced in [126]) should be written up [127] soon. For the derivation of formulas and exponents for critical percolation using SLE6, see [124, 92, 128]. 11 What is missing 11.1 A list of ideas We have listed at the end of each chapter a ”
Critical percolation exploration path and SLE6: a proof of convergence (2006, background): “tain critical exponents [34], of Watt’s crossing formula [14] and Schramm’s percolation formula [28], and the derivation of the full scaling limit [8,9].”
SLE for theoretical physicists (2005, background): “Given a curve γ connecting two points r1 and r2 on the boundary of a domain D, what is the probability that it passes to the left of a given interior point? This is not a question which is natural in conventional approaches to critical behaviour, but which is very simply answered within SLE[25].”
One-Arm Exponent for Critical 2D Percolation (2001, background): “48 ≤ P dist(Q,0) <r ≤ cr5/48. This theorem resembles instatement andinproofthe determination ofthe Brownian motion exponents carried out in [14]. Also related to this theorem is the calculation in [21] of the probability of an event for the scaling limit. We will assume thatthe reader is familiar”
Conformally Invariant Processes in the Plane (2005, unknown): “O. Schramm (2001), A percolation formula, Electron.”