Idleness Functions for Ollivier-Ricci Curvature on Hypergraphs

Let $\mathcal H=(V,E)$ be a locally finite simple hypergraph, equip $V$ with the hyperpath metric, and consider the equal-edges random walk \cite{CoupetteEtAl2023}. For adjacent vertices $x$ and $y$, we prove that the idleness function $α\mapstoκ_α^{\mathcal H}(x,y)$ is piecewise affine and with no