Differentiability and strict convexity of the limit shape in directed FPP

Establish differentiability and strict convexity of the time constant (limit shape) ℓ on the set of directions 𝒰 for planar directed edge‑weight first passage percolation on ℤ² under Assumption 2.1 (i.i.d. non‑constant weights with finite 2+ε moments). This would remove the current conditionality of results that assume ℓ is differentiable and strictly convex and extend their validity beyond the integrable weight distributions treated in the paper.

Background

The paper’s existence and structural results for the Busemann process are proved under the assumption that the limit shape ℓ is differentiable and strictly convex. While these properties can be directly verified in several integrable models (e.g., Bernoulli–exponential and Bernoulli–geometric weights), they are not presently known in general for directed edge‑weight FPP on ℤ².

Removing this assumption is important for establishing unconditional versions of the Busemann process results (existence, ergodicity, monotonicity, and continuity), as well as for deriving geodesic properties without relying on model‑specific integrability.

References

This remains unproven in any level of generality, but can be directly verified for the models we treat in the remainder of the paper, where explicit representations exist.

— The Busemann Process and Steep Highways in Directed First Passage Percolation  (2510.19159 - McKeown, 22 Oct 2025) in Section 2.2 (Existence of the Busemann process)

In Regimes \rm V_E and \rm V_H, does an almost-sure shape theorem hold, i.e.,\ is there a norm \nu on \dRd with n{-1}T(\mvzero,\lceil n\mvx\rceil)\to\nu(\mvx) a.s.? Subadditivity guarantees a deterministic finite limit along fixed rays under stronger moment conditions (e.g., on V{-1} and \go). However, establishing a full shape theorem requires further control to ensure uniform convergence. If a limit shape exists, the simulations of Figure~\ref{fig:sim-V} suggest it is close to a Euclidean ball in \rm V_E but visibly distorted in \rm V_H; it would be interesting to determine whether it can fail to be strictly convex.

— Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment  (2609.20422 - Chatterjee et al., 17 Sep 2026) in Item 3, Subsection 1.5.1 (Sharp asymptotics)