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Publication 66

Γ-Convergence of Perimeter on Random Geometric Graphs

Authors:

CMUNicolás García Trillos
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213


CMUDejan Slepčev
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213


Abstract:
We study the perimeter on the sets $V_n$ of $n$ random, uniformly distributed points in a Euclidean domain. To define the notion of perimeter on a finite set of points, the set is considered as a weighted graph, where the weights are assigned to edges connecting pairs of points based on their distances. The perimeter of $A_n \subset V_n$ is defined by summing the weights of edges between $A_n$ and $V_n \backslash A_n$. We investigate under which choice of weights do the functionals which assign the graph perimeter converge to the perimeter in the Euclidean space as the number of points $n$ goes to infinity. In particular, for $\varepsilon(n)$ such that significant weight is given to edges of length up to $\varepsilon(n)$ we investigate under which scaling of $\varepsilon$ on $n$ does the convergence hold. We consider this question in the setting of $\Gamma$-convergence and consider it for total-variation functional on graphs (which extends the notion of the perimeter).
Get the paper in its entirety
13-CNA-024.pdf

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