Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

Abstract One of the main obstacles regarding Bakry–Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry–Emery curvature everywhere but on the exception set.