Harmonic explorer and its convergence to SLE4

By Oded Schramm, Scott Sheffield. 2003

144 citations recorded by the source corpus.

The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE 4 as the grid gets finer.

Subfields: Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/9912d78437fd9b8137274e7e9da24ecb369adce1

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Citations in the corpus

The scaling limits of planar LERW in finitely connected domains. (2006, background): “ts of UST (uniform spanning tree) Peano curve. Smirnov proved in [18] that chordal SLE 6 is the scaling limit of critical site percolation on the triangular lattice. And Schramm and Sheﬃeld proved in [17] that the harmonic explorer converges to chordal SLE 4. In [9], SLE 8/3 is proved to have the re”
Critical percolation exploration path and SLE6: a proof of convergence (2006, background): “We note (see Remark 5.2 below for more discussion) that our choice of stopping times is closer in spirit than is the choice in [31,32] to the proofs of convergence of the loop erased random walk to SLE2 [22] and of the harmonic explorer to SLE4 [ 29 ].”
Towards conformal invariance of 2D lattice models (2007, background): “In [35] Oded Schramm and Scott Sheffield introduced this curve with a nicer “Harmonic Explorer” definition, and utilizing the mentioned observable showed that it indeed converges to SLE (4).”
The dimension of the SLE curves (2002, methodology): “e proved that SLE 2 and SLE 8 are the respective scaling limits of planar loop-erased random walks and uniform Peano curves. 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 i”
Contour lines of the two-dimensional discrete Gaussian free field (2006, background): “... from the statistical physics literature are conjectured to have forms of SLE as scaling limits, rigorous proofs have thus far appeared only for a few cases: site percolation cluster boundaries on the hexagonal lattice (SLE(6), [Sm]; see also [CN]), branches (loop-erased random walk) and outer bo”
Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, methodology): “Situation is somewhat easier with the observables for the Harmonic Explorer and Discrete Gaussian Free Field, as discussed by Schramm and Sheffield [SS05, SS09] – both are harmonic and solving Dirichlet problem with bounded boundary values, so the generalization from the original triangular lattice ”
Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “These models are the harmonic explorer [80] and the interface of the discrete Gaussian free field [81].”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “We have already mentioned that Smirnov [40], [41], [42] has proved this conjecture for some important models (percolation, Ising model; see also [18], [33], [34] for some other cases).”