One-Arm Exponent for Critical 2D Percolation

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2001

235 citations recorded by the source corpus.

The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R$ to the power $5/48$ as $R$ goes to infinity.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/799e8bc9f628f726fd85b2af55f25725689b599a

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In Condensed Matter Physics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

SLE for theoretical physicists (2005, unknown): “A more careful argument [32] confirms this.”
Critical percolation exploration path and SLE6: a proof of convergence (2006, methodology): “We start with two lemmas from [8,9], which are consequences of [2] and of standard bounds on the probability of events corresponding to having a certain number of disjoint monochromatic crossings of an annulus (see Lemma 5 of [17], Appendix A of [21], and also [3]).”
Two-Dimensional Critical Percolation: The Full Scaling Limit (2006, background): “The work of Schramm [32] and Smirnov [36] has identified the scaling limit of a certain percolation interface with SLE6, providing, along with the work of Lawler-Schramm-Werner [25, 26] and Smirnov-Werner [40], a confirmation of many results in the physics literature, as well as some new results.”
Lectures on two-dimensional critical percolation (2007, methodology): “nvergenceto SLE(6), we follow the strategy outlined by Smirnov in [41] (see also our (we=Lawler-Schramm-Werner) paper on loop-erased random walks [26]). • We will follow some SLE(6) computations from [24, 25]. • For how to deduce the critical exponents for critical percolation from the above, we fol”
The Random-Cluster Model (2002, background): “It is possible to perform calculations on stochastic Löwner evolutions, and in particular to confirm, [104, 134], the values of many critical exponents associated with percolation (or, at least, site percolation on the triangular lattice).”
Quantitative noise sensitivity and exceptional times for percolation (2005, background): “For example, for critical site percolation on the triangular grid, it was established in [23] that α(R) = R−5/48+o(1) as R→∞ .”
Random planar curves and Schramm-Loewner evolutions (2003, methodology): “e reader can ﬁnd more detailed proofs. It is possible to compute analogous exponents for chordal SLE. These “half-plane exponents” are determined in [86, 88]. Other important exponents are derived in [118, 92, 15]. As in this chapter, the exponents appear always as leading eigenvalues of some diﬀere”
Near-critical percolation in two dimensions (2007, methodology): “The only result proved so far concerns the case j = 2: as shown in [36], the monochromatic 2-arm exponent can be expressed as the leading eigenvalue of some (complicated) differential operator.”
The Alexander-Orbach conjecture holds in high dimensions (2008, methodology): “This exponent takes the value 48/5 in the two dimensional triangular grid, as shown by Lawler, Schramm and Werner [41] and Smirnov [53].”