Lipschitz-continuity of time constant in generalized First-passage percolation

In this article, we consider a generalized First-passage percolation model, where each edge in $Z^d$ is independently assigned an infinite weight with probability $1-p$, and a random finite weight otherwise. The existence and positivity of the time constant have been established in Cerf and Théret (2016). Recently, using sophisticated multi-scale renormalizations, Cerf and Dembin (2022) proved that the time constant of chemical distance in super-critical percolation is Lipschitz continuous. In this work, we propose a different approach leveraging lattice animal theory and a simple one-step renormalization with the aid of Russo’s formula, to show the Lipschitz continuity of the time constant in generalized First-passage percolation.