arXiv subjects

Christian Benes

Publications and source records attributed to Christian Benes.

4 recordsLinked to original sources

arXiv2020

Rates of Convergence for the Planar Discrete Green's Function in Pacman Domains

We obtain upper bounds for the rates of convergence for the simple random walk Green's function in the domains $D_α= D_α(n)=\{re^{iθ}\in \mathbb{C}:0 <θ<2π-α, 0<r<2n\}-z_0,$ where $z_0\in\mathbb{Z}^2$ is a point closest to $ne^{i(π-α/2)}$. The rate depends on the angle of the wedge and is what was suggested by the sharpest available results in the extreme cases $α=0$ and $α=π$. Our proof uses the KMT coupling between random walk and Brownian motion.

math.PR↗

arXiv2015

Scaling limit of the loop-erased random walk Green's function

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the SLE(2) Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a "spinor" observable for random walk started from one of the vertices of the marked edge.

math.PR↗

arXiv2013

On the rate of convergence of loop-erased random walk to SLE(2)

We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE(2), we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.

math.PR↗

arXiv2006

Some Estimates for Planar Random Walk and Brownian Motion

The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for coupling the two processes, as well as solutions to the discrete Dirichlet problem in various domains which, to the author's knowledge, have not been published anywhere. The main focus is on the two-dimensional processes.

math.PR↗