Differentiability of limit shapes in continuous first passage percolation models

TL;DR

This paper introduces a general framework for continuous first passage percolation models, demonstrating conditions under which the limit shape is differentiable, with applications to Riemannian and Poissonian point-based models.

Contribution

It provides a unifying framework for analyzing differentiability of limit shapes in continuous percolation models, extending previous results to new classes of models.

Findings

Limit shape exists in the proposed models.

Under certain conditions, the limit shape is differentiable.

Examples include Riemannian and Poissonian point models.

Abstract

We introduce and study a class of abstract continuous action minimization problems that generalize continuous first and last passage percolation. In this class of models a limit shape exists. Our main result provides a framework under which that limit shape can be shown to be differentiable. We then describe examples of continuous first passage percolation models that fit into this framework. The first example is of a family of Riemannian first passage percolation models and the second is a discrete time model based on Poissonian points.