An asymptotic expansion for the discrete harmonic potential
Gady Kozma,E. Schreiber +1 more
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In this article, the harmonic potential of a random walk has been shown to have arbitrary precision asymptotics, and two algorithms that allow to obtain arbitrary precision harmonic potentials for random walks are given.Abstract:
We give two algorithms that allow to get arbitrary precision asymptotics for the harmonic potential of a random walk.read more
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Annealed estimates on the Green function
Daniel Marahrens,Felix Otto +1 more
TL;DR: In this paper, the authors considered a random, uniformly elliptic coefficient field (a(x)) on the Z-dimensional integer lattice and showed that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds.
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The scaling limit of loop-erased random walk in three dimensions
TL;DR: In this article, it was shown that the scaling limit exists and is invariant under dilations and rotations, and some tools that might be useful to show universality were presented.
Journal ArticleDOI
Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
TL;DR: In this article, the authors studied the local structure of the extremal process associated with the Discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of bounded open planar domains subject to mild regularity conditions on the boundary.
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Logarithmic fluctuations for internal DLA
TL;DR: In this paper, it was shown that the discrepancy between A(n) and the disk is at most logarithmic in the radius, i.e., there is an absolute constant C such that with probability 1, Br C log r A(r 2 ) Br+C log r for all suciently large r:
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Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field
TL;DR: In this paper, the authors studied the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidated how the conformal symmetries manifest themselves in the scaling limit.
References
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Introduction to Fourier Analysis on Euclidean Spaces.
Elias M. Stein,Guido Weiss +1 more
TL;DR: In this paper, the authors present a unified treatment of basic topics that arise in Fourier analysis, and illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations.
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XXII.—Random Paths in Two and Three Dimensions.
W. H. McCrea,F. J. W. Whipple +1 more
TL;DR: In this article, a rectangular lattice is given and a particle P moves from one lattice-point to another in such a way that, when it is at any interior point, it is equally likely to move to any of its four neighbouring points.
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Green's Functions for Random Walks on ZN
TL;DR: In this paper, the Martin boundary of a random walk has been shown to behave like a non-harmonic function when the second moment is bounded by a constant number of points.
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Potential kernel for two-dimensional random walk
Yasunari Fukai,Kôhei Uchiyama +1 more
TL;DR: In this article, it was shown that the potential kernel of a recurrent, aperiodic random walk on the integer lattice admits an asymptotic expansion of the form $$(2 \pi \sqrt{|Q|})^{-1} \ln Q(x_2, -x_1) + \const + |x|^{- 1} U_1 (\omega^x) + | x|-2}U_2 ( \omega + \dots) +