Nonconvex Carnot Limit Shapes for Branching Random Walks on the Heisenberg Group
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 n2. 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 Jt(ω)≤ tlogρ for every t∈[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≥4, 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 Γρ is nondecreasing and concave in ρ, equals 1/4 for ρ≥4, and satisfies Γρ=(logρ)/π−(logρ)2/(4π) +O((logρ)3) as ρ↓1. For fixed 1<ρ<4, we obtain MnZ=Γρn2+O(n3/2√(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 Γ4n2−MnZ has order nloglog n in probability.
Related
- On the critical branching random walk I: branching capacity and visiting probability
- Capacities and Hitting Times for Simple and (critical) Branching Random Walk Sausages
- Domains of Attraction for Quasi-Stationary Distributions of Branching Processes in Random Environments
- High-dimensional critical percolation and critical Branching random walks are intersection equivalent
- Recurrence and capacity of stable branching random walks
BibTeX
@misc{hu2026nonconvex,
title = {{Nonconvex Carnot Limit Shapes for Branching Random Walks on the Heisenberg Group}},
author = {Bohan Hu and Jungang Li and Wen Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23162},
eprint = {2610.23162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23162},
}
RIS
TY - UNPB
TI - Nonconvex Carnot Limit Shapes for Branching Random Walks on the Heisenberg Group
AU - Bohan Hu
AU - Jungang Li
AU - Wen Sun
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23162
UR - https://arxiv.org/abs/2610.23162
AB - 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)≤ tlogρ for every t∈[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≥4, 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≥4, and satisfies Γ_ρ=(logρ)/π−(logρ)^2/(4π) +O((logρ)^3) as ρ↓1. For fixed 1<ρ<4, we obtain M_n^Z=Γ_ρn^2+O(n^3/2√(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 nloglog n in probability.
ER -