Nonconvex Carnot Limit Shapes for Branching Random Walks on the Heisenberg Group

Source abstract

We study a supercritical branching random walk on the discrete Heisenberg group driven by the four standard horizontal generators. Horizontal position has scale $n$, whereas the central coordinate records discrete area and has scale $n^2$. On the Galton-Watson survival event, we prove almost sure Hausdorff convergence, under Carnot scaling, of both the rescaled particle positions and the collections of enhanced ancestral paths. The path limit consists of area lifts of horizontal paths $ω$ satisfying $J_t(ω)\le t\logρ$ for every $t\in[0,1]$, where $ρ$ is the offspring mean. In contrast to the classical Euclidean case, the deterministic limit shape is nonconvex for every $ρ>1$. We identify its horizontal projection and interval vertical fibers, determine the saturated shape for $ρ\ge4$, and characterize the origin fiber below saturation by Wulff duality. A hyperbolic representation gives an elliptic integral formula for the Wulff area. The maximal central coordinate $Γ_ρ$ is nondecreasing and concave in $ρ$, equals $1/4$ for $ρ\ge4$, and satisfies $Γ_ρ=(\logρ)/π-(\logρ)^2/(4π) +O((\logρ)^3)$ as $ρ\downarrow1$. For fixed $1<ρ<4$, we obtain $M_n^Z=Γ_ρn^2+O(n^{3/2}\sqrt{\log n})$ almost surely. Above saturation the first correction is a random linear term; at $ρ=4$, under a finite second moment assumption, the deficit $Γ_4n^2-M_n^Z$ has order $n\log\log n$ in probability.