TY - JOUR

T1 - The First Order Correction to the Exit Distribution for Some Random Walks

AU - Kennedy, Tom

N1 - Funding Information:
This research was partially supported by NSF Grant DMS-1500850. An allocation of computer time from the UA Research Computing High Performance Computing (HPC) and High Throughput Computing (HTC) at the University of Arizona is gratefully acknowledged. The author thanks Jianping Jiang for many stimulating conversations about this research.
Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - We study three different random walk models on several two-dimensional lattices by Monte Carlo simulations. One is the usual nearest neighbor random walk. Another is the nearest neighbor random walk which is not allowed to backtrack. The final model is the smart kinetic walk. For all three of these models the distribution of the point where the walk exits a simply connected domain D in the plane converges weakly to harmonic measure on ∂D as the lattice spacing δ→ 0. Let ω(0 , · ; D) be harmonic measure for D, and let ωδ(0 , · ; D) be the discrete harmonic measure for one of the random walk models. Our definition of the random walk models is unusual in that we average over the orientation of the lattice with respect to the domain. We are interested in the limit of (ωδ(0 , · ; D) - ω(0 , · ; D)) / δ. Our Monte Carlo simulations of the three models lead to the conjecture that this limit equals cM,LρD(z) times Lebesgue measure with respect to arc length along the boundary, where the function ρD(z) depends on the domain, but not on the model or lattice, and the constant cM , L depends on the model and on the lattice, but not on the domain. So there is a form of universality for this first order correction. We also give an explicit formula for the conjectured density ρD.

AB - We study three different random walk models on several two-dimensional lattices by Monte Carlo simulations. One is the usual nearest neighbor random walk. Another is the nearest neighbor random walk which is not allowed to backtrack. The final model is the smart kinetic walk. For all three of these models the distribution of the point where the walk exits a simply connected domain D in the plane converges weakly to harmonic measure on ∂D as the lattice spacing δ→ 0. Let ω(0 , · ; D) be harmonic measure for D, and let ωδ(0 , · ; D) be the discrete harmonic measure for one of the random walk models. Our definition of the random walk models is unusual in that we average over the orientation of the lattice with respect to the domain. We are interested in the limit of (ωδ(0 , · ; D) - ω(0 , · ; D)) / δ. Our Monte Carlo simulations of the three models lead to the conjecture that this limit equals cM,LρD(z) times Lebesgue measure with respect to arc length along the boundary, where the function ρD(z) depends on the domain, but not on the model or lattice, and the constant cM , L depends on the model and on the lattice, but not on the domain. So there is a form of universality for this first order correction. We also give an explicit formula for the conjectured density ρD.

KW - Exit distribution

KW - First order correction

KW - Harmonic measure

KW - Random walk

KW - Smart kinetic walk

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U2 - 10.1007/s10955-016-1534-8

DO - 10.1007/s10955-016-1534-8

M3 - Article

AN - SCOPUS:84969220165

VL - 164

SP - 174

EP - 189

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -