论文标题
Dirichlet Energy在泊松点云上的渐近行为
Asymptotic behavior of the Dirichlet energy on Poisson point clouds
论文作者
论文摘要
我们证明，在平面泊松云上定义的函数的二次相互作用，并考虑到距离的位置对，直至一定的（较大的）阈值，几乎可以通过确定性常数的多个dirichlet能量来肯定会近似。这是通过缩放泊松云和相应的能量并计算紧凑的离散到核电限制来实现的。为了避免泊松云的特殊区域的效果，堆积位点或“断开位点”，必须给出适当的“粗粒”函数收敛概念。
We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with "disconnected sites", a suitable "coarse-grained" notion of convergence of functions defined on scaled Poisson clouds must be given.
