Potential kernel for two-dimensional random walk

Abstract

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice ‫ޚ‬ 2 admits an asymptotic expansion of the form

On the accuracy of finite difference methods for elliptic problems with interfaces

“…Theorem 2.1 and the lemmas also hold with periodic or Neumann boundary conditions, rather than Dirichlet, as discussed below. For the discrete Dirichlet problem (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18), it is well known that the maximum of v can be estimated by the maximum of the right side, using the discrete maximum principle, but (2-21) is sharper in dependence on F k .…”

Abelian Sandpiles and the Harmonic Model

“…2 However, there is a distinguished (or fundamental) solution w (d) of (2.5) which has a deep probabilistic meaning: it is a certain multiple of the lattice Green's function of the symmetric nearest-neighbour random walk on Z d (cf. [6,12,25,27]). …”

Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field

“…has a positive density on [j − 1, j) × W δ , which is uniformly bounded from below with a bound not depending on either j or K, and that the Lévy distance between µ j g and µ j c is bounded from above by ε K → K→∞ 0, due to (14). Since [j − 1, j) × W δ is a bounded subset of R 3 , an elementary coupling (see, e.g., [6, Theorem 1.2]; the analog for one-dimensional couplings is easy to check) yields a coupling satisfying the analog of (134), but restricted to [j − 1, j).…”