Continuity and differentiability of percolation transition probabilities

Establish whether the probabilities that a vertex belongs to an infinite infected component or an infinite healthy component under the upper invariant measure are continuous at their respective thresholds, and determine whether the probability of belonging to an infinite healthy component has a continuous derivative at \(\lambda=\lambda_{p^\complement}\) while the corresponding infected-component probability does not have a continuous derivative at \(\lambda=\lambda_p\).

Background

The paper presents simulations of the probabilities that a vertex belongs to an infinite connected component of infected vertices or healthy vertices. The numerical evidence suggests continuous transitions, but the authors do not claim a proof and note a possible difference in differentiability between the two transition points.

The unresolved questions concern the regularity of the percolation probabilities at $\lambda_p$ and $\lambda_{p^\complement}$, including whether the healthy-component probability may have a continuous derivative while the infected-component probability may exhibit a derivative discontinuity.