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Recursive Paintboxes and the Martin Boundary of the Hoffman Rooted-Tree Graph

  • Shengjun Zhang
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Abstract

We determine the Doob-Martin boundary of Hoffman's leaf-grafting graph on finite unlabelled non-plane rooted trees. Its full and minimal boundaries coincide and are parametrized by deterministic recursive paintboxes, identified when their finite sampling laws agree. Every central measure is a unique mixture of the corresponding extremal laws, and its limiting boundary point generates the completed tail field. We also show that the boundary is homeomorphic to the space of unordered root masses marked by child boundary classes. For the recursive Ewens family, we obtain the unique extremal decomposition from independent Poisson-Dirichlet splits, including the uniform recursive-tree and rooted-tree Plancherel cases.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026recursive,
  title = {{Recursive Paintboxes and the Martin Boundary of the Hoffman Rooted-Tree Graph}},
  author = {Shengjun Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30135},
  eprint = {2609.30135},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30135},
}
RIS
TY  - UNPB
TI  - Recursive Paintboxes and the Martin Boundary of the Hoffman Rooted-Tree Graph
AU  - Shengjun Zhang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30135
UR  - https://arxiv.org/abs/2609.30135
AB  - We determine the Doob-Martin boundary of Hoffman's leaf-grafting graph on finite unlabelled non-plane rooted trees. Its full and minimal boundaries coincide and are parametrized by deterministic recursive paintboxes, identified when their finite sampling laws agree. Every central measure is a unique mixture of the corresponding extremal laws, and its limiting boundary point generates the completed tail field. We also show that the boundary is homeomorphic to the space of unordered root masses marked by child boundary classes. For the recursive Ewens family, we obtain the unique extremal decomposition from independent Poisson-Dirichlet splits, including the uniform recursive-tree and rooted-tree Plancherel cases.
ER  -