Values of Brownian intersection exponents III: Two-sided exponents

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2000

194 citations recorded by the source corpus.

Abstract This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. For instance, it is proven that the exponent ξ(3,3) describing the asymptotic decay of the probability of non-intersection between two packs of three independent planar Brownian motions each is (73−2 73 )/12 . More generally, the values of ξ(w1,…,wk) and ξ (w 1 ′,…,w k ′) are determined for all k⩾2, w1,w2⩾1, w3,…,wk∈[0,∞) and all w′1,…,w′k∈[0,∞). The proof relies on the results derived in our first two papers and applies the same general methods. We first find the two-sided exponents for the stochastic Loewner evolution processes in a half-plane, from which the Brownian intersection exponents are determined via a universality argument.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/536a36574574c0594cf9d5bbc3915f531c2f2d29

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Citations in the corpus

The scaling limits of planar LERW in finitely connected domains. (2006, unknown): “The Annals of Probability 2008, Vol. 36, No. 2, 467–529 DOI: 10.1214/07-AOP342 © Institute of Mathematical Statistics, 2008”
Liouville quantum gravity and KPZ (2008, methodology): “It was conjectured in [DK88b] and rigorously derived in a celebrated series of papers by Lawler, Schramm, and Werner using the Schramm-Loewner evolution with κ = 6 [LSW01a, LSW01b, LSW02 ]:”
Random planar curves and Schramm-Loewner evolutions (2003, methodology): “er model (that should be closely related to critical percolation scaling limits) ﬁrst appeared in [97]. The mathematical derivation of the value of the exponents was performed in the series of papers [86, 87, 88, 89]. The properties of SLE that were later derived in [95] enable to shorten some parts”
CRITICAL EXPONENTS FOR TWO-DIMENSIONAL PERCOLATION (2001, background): “This is the so-called correlation length corresponding to the “typical radius” of a finite cluster.”
Basic properties of SLE (2001, background): “In the series of papers [LSW01a, LSW01b, LSW02 ], a number of other properties of SLE have been studied.”
One-Arm Exponent for Critical 2D Percolation (2001, methodology): “Loosely speaking, this lemma says that the two-arms half plane exponent is 1. Note that the half-plane exponents for SLE6 were calculated in [13, 15 ].”
Excursion decompositions for SLE and Watts' crossing formula (2004, background): “for some u1 17 ]).”
Gaussian multiplicative chaos and applications: A review (2013, methodology): “For instance, Duplantier [52] have used these techniques to conjecture the exact values of the Brownian intersection exponents, which were finally rigorously derived in [102, 103, 104] via Schramm-Loewner-Evolution (SLE) techniques.”