The scaling limits of planar LERW in finitely connected domains.
By Dapeng Zhan. 2006
76 citations recorded by the source corpus.
We define a family of stochastic Loewner evolution-type processes in finitely connected domains, which are called continuous LERW (loop-erased random walk). A continuous LERW describes a random curve in a finitely connected domain that starts from a prime end and ends at a certain target set, which could be an interior point, or a prime end, or a side arc. It is defined using the usual chordal Loewner equation with the driving function being √2B(t) plus a drift term. The distributions of continuous LERW are conformally invariant. A continuous LERW preserves a family of local martingales, which are composed of generalized Poisson kernels, normalized by their behaviors near the target set. These local martingales resemble the discrete martingales preserved by the corresponding LERW on the discrete approximation of the domain. For all kinds of targets, if the domain satisfies certain boundary conditions, we use these martingales to prove that when the mesh of the discrete approximation is small enough, the continuous LERW and the corresponding discrete LERW can be coupled together, such that after suitable reparametrization, with probability close to 1, the two curves are uniformly close to each other.
Subfields: Mathematical Physics, Molecular Biology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/ba0de1c290f138517e42bf713841ff00398f26e5
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Citations in the corpus
- Duality of chordal SLE (2007, background): “…their scaling limits when the mesh of the grid tends to 0, e.g., the convergence of critical percolation on triangular lattice to SLE(6) [16], loop-erased random walk (LERW) to SLE(2) [9,18], uniform spanning tree (UST) Peano curve to SLE(8) [9], Gaussian free field contour line to SLE(4) [13],…”
- Reversibility of chordal SLE (2008, background): “1) in [17] for z ∈ R\ [cH , dH ], and the fact that φH is increasing on (−∞, aH) and (bH ,∞), and maps them to (−∞, cH) and (dH ,∞), respectively.”