Classify maximizing paths below saturation

Classify the paths maximizing the central front Γρ for each offspring mean 1 < ρ < 4, including their possible contact sets with the intermediate action constraints.

Background

For the Heisenberg branching random walk with offspring mean 1 < ρ < 4, the central front is the variational quantity Γρ = supω∈Cρ A(ω), where Cρ consists of absolutely continuous horizontal paths whose accumulated action satisfies Jt(ω) ≤ t log ρ for every t ∈ [0,1]. The paper establishes the value’s asymptotic behavior near weak branching and proves that intermediate-time constraints are genuinely active, but it does not classify all maximizing paths in the sub-saturation regime.

The unresolved classification problem includes determining the contact set {t ∈ [0,1] : Jt(ω) = t log ρ} for maximizers and deciding whether maximizers are unique up to the symmetries of the limit shape stated in (5.7).