On the scaling limit of planar self-avoiding walk

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2002

255 citations recorded by the source corpus.

A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the scaling limit of the uniform measures on these objects. The conjectures are based on recent results on the stochastic Loewner evolution and non-disconnecting Brownian motions. New heuristic derivations are given for the critical exponents for SAWs and SAPs.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4f6c9a4dcf7247544c98a9fa6711b20c7af4ef1b

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Citations in the corpus

The dimension of the SLE curves (2002, result): “Convergence of the “harmonic explorer” was obtained by Schramm and Sheffield [ 16], and there is also strong evidence [ 13 ] that the infinite self-avoiding walk in the half-plane is related to SLE 8/3 .”
THE SELF-AVOIDING WALK (2006, background): “In particular, Lawler, Schramm and Werner among others conjectured that SLE8/3 in a simple connected domain D $ C, provides the scaling limit for the self-avoiding walk in two dimensions (see [17] and [19] for more details).”
Hausdorff dimensions for SLE6 (2002, background): “We should also mention here that SLE8/3 is the natural candidate for the scaling limit of self-avoiding walks [12] and therefore also an interesting object.”
The configurational measure on mutually avoiding SLE paths (2006, background): “For λ = 0, this is equivalent to very difficult conjectures about self-avoiding walks; see [12].”
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$ (2010, background): “3 of [5], precise behaviors are conjectured for the number of self-avoiding walks between two points on the boundary of a domain.”
Recent advances in percolation theory and its applications (2015, background): “For example, the external perimeter of a percolation cluster in the continuum limit is believed to be described by SLE8/3 which is also expected to describe the scaling limit of planar self-avoiding random walk (SAW), although there is no complete mathematical proof yet [261]—see also Fig.”
Conformal restriction: The chordal case (2002, background): “…by adding to the chordal SLE (cid:20) curve certain Brownian bubbles with intensity (cid:21) , where (cid:11) , (cid:21) and (cid:20) are related by (4) For all (cid:11) (cid:21) 5 = 8, the dimension of the boundary of K de(cid:12)ned under P (cid:11) is almost surely 4 = 3 and locally \looks…”