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.