Poisson-Daunay三角剖分上的视线Cox渗流

Harmonic deformation of Delaunay triangulations

We construct harmonic functions on random graphs given by Delaunay triangulations of ergodic point processes as the limit of the zero-temperature harness process.

Quenched invariance principle for random walks on Delaunay triangulations

We consider simple random walks on Delaunay triangulations generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on the point processes, we show that the random walk satisfies an almost sure (or quenched) invariance principle. This invariance principle holds for point processes which have clustering or repulsiveness properties including Poisson point processes, Mat{é}rn cluster and Mat{é}rn hardcore processes. The method relies on the decomposition of the process into a martingale part and a corrector which is proved to be negligible at the diffusive scale.

Stretch Factor of Long Paths in a planar Poisson-Delaunay Triangulation

Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We provide a first non-trivial lower bound for the distance between the expected length of the shortest path between $(0,0)$ and $(1,0)$ in the Delaunay triangulation associated with $X$ when the intensity of $X_n$ goes to infinity. Experimental values indicate that the correct value is about 1.04. We also prove that the expected number of Delaunay edges crossed by the line segment $[(0,0),(1,0)]$ is equivalent to $2.16\sqrt{n}$ and that the expected length of a particular path converges to 1.18 gi…

Coalescence of Euclidean geodesics on the Poisson-Delaunay triangulation

Let us consider Euclidean first-passage percolation on the Poisson-Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an adapted Burton-Keane argument and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesi…

二维空间中基于德劳内三角剖分的分数布朗运动平方增量和的中心极限定理

Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$

研究在单位正方形内由泊松点过程生成的点集，基于德劳内三角剖分考虑归一化增量和，推导当点密度趋于无穷时的中心极限定理。

Random Delaunay triangulations, the Thurston-Andreev theorem, and metric uniformization

In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Delaunay triangulation of a Riemannian surface to form an energy measuring how ``uniform'' a metric is, see also math.DG/0010316.