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Hyperbolic radial spanning tree

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  • Coupier, David
  • Flammant, Lucas
  • Tran, Viet Chi

Abstract

We define and analyze an extension to the d-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. To obtain results for any dimension, a completely new approach is considered here. Our strategy mainly involves the two following ingredients, that rely on the hyperbolic Directed Spanning Forest (DSF) introduced and studied in Flammant (2019). First, the hyperbolic metric allows us to obtain fine control of the branches’ fluctuations in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF and conclude thanks to the precise estimates mentioned before.

Suggested Citation

  • Coupier, David & Flammant, Lucas & Tran, Viet Chi, 2024. "Hyperbolic radial spanning tree," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    • Handle: RePEc:eee:spapps:v:172:y:2024:i:c:s0304414924000243
    DOI: 10.1016/j.spa.2024.104318
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    References listed on IDEAS

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    1. Marián Boguñá & Fragkiskos Papadopoulos & Dmitri Krioukov, 2010. "Sustaining the Internet with hyperbolic mapping," Nature Communications, Nature, vol. 1(1), pages 1-8, December.
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