The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors of the current endpoint of the walk. In the full-plane a nearest neighbor is allowable if it has not been visited before and there is a path from the nearest neighbor to infinity through sites that have not been visited before. It is well known that on the hexagonal lattice the SKSAW in a bounded domain between two boundary points is equivalent to an interface in critical percolation, and hence its scaling limit is the chordal Schramm–Loewner evolution with $$\kappa =6$$κ=6 (SLE$$_6$$6). Like SLE there are variants of the SKSAW depending on the domain and the initial and terminal points. On the hexagonal lattice these variants have been shown to converge to the corresponding version of SLE$$_6$$6. It is believed that the scaling limit of all these variants on any regular lattice is the corresponding version of SLE$$_6$$6. We test this conjecture for the square lattice by simulating the SKSAW in the full-plane and find excellent agreement with the predictions of full-plane SLE$$_6$$6.

Original languageEnglish (US)
Pages (from-to)302-320
Number of pages19
JournalJournal of Statistical Physics
Volume160
Issue number2
DOIs
StatePublished - Apr 25 2015

Fingerprint

Self-avoiding Walk
Kinetics
Hexagonal Lattice
Nearest Neighbor
Scaling Limit
kinetics
Walk
scaling
Intersect
random walk
Square Lattice
infinity
Bounded Domain
Random walk
Choose
Infinity
Converge
Path
Prediction
predictions

Keywords

  • Laplacian random walk
  • Percolation explorer
  • Schramm–Loewner evolution
  • Smart kinetic walk

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution. / Kennedy, Thomas G.

In: Journal of Statistical Physics, Vol. 160, No. 2, 25.04.2015, p. 302-320.

Research output: Contribution to journalArticle

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