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Gibbs Partitions and Lattice Paths

Authors Niccolò Bosio , Markus Kuba , Benedikt Stufler

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Probability and statistics
Keywords
  • Gibbs partitions
  • composition schemes
  • lattice paths
  • random walks
  • condensation

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Abstract

We study a Gibbs partition model (composition scheme) under a new condition on the component weights, leading to a previously unobserved regime for the number of components. We establish a condensation phenomenon producing a unique giant component, and prove a Cox process limit describing a sublinear power-law growth of sizes of non-maximal components. Our results are motivated by applications to lattice paths and random walks, including simple random walks in the cube, Delannoy paths, pairs of Dyck bridges, urn models and card guessing games.

Cite As Get BibTex

Niccolò Bosio, Markus Kuba, and Benedikt Stufler. Gibbs Partitions and Lattice Paths. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 10:1-10:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.10

Author Details

Niccolò Bosio
  • Vienna University of Technology, Austria
Markus Kuba
  • University of Applied Sciences - Technikum Wien, Austria
Benedikt Stufler
  • Vienna University of Technology, Austria

Funding

This research was funded in part by the Austrian Science Fund (FWF) 10.55776/F1002 and 10.55776/PAT6732623. For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.

Acknowledgements

The authors thank Alois Panholzer for several discussions about combinatorial applications of our results, his generosity and his interest in this work. The second author also thanks Cyril Banderier for interesting discussions concerning the new component weights treated in this article, their connection to families of lattice paths and the new limit laws for composition schemes.

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