$\Gamma$-convergence of graph Ginzburg-Landau functionals

Abstract

We study $\Gamma$-convergence of graph-based Ginzburg--Landau functionals, both the limit for zero diffusive interface parameter $\varepsilon \to 0$ and the limit for infinite nodes in the graph $m \to \infty$. For general graphs we prove that in the limit $\varepsilon \to 0$ the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg--Landau functional. For both functionals we also study the simultaneous limit $\varepsilon \to 0$ and $m \to \infty$, by expressing $\varepsilon$ as a power of $m$ and taking $m \to \infty$. Finally we investigate the continuum limit for a nonlocal means-type functional on a completely connected graph.