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Faster Triangulation Mixing via Transport Flows

Authors Vedat Levi Alev , Daniel Frishberg , Michail Sarantis , Prasad Tetali

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ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • triangulations
  • mixing time
  • log-Sobolev inequality
  • spectral gap
  • Markov chain
  • random walk
  • MCMC
  • transport flow
  • multicommodity flow

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Abstract

We prove an Õ(n²) bound for the relaxation time and the log-Sobolev time (inverse log-Sobolev constant) of the classical triangulation flip chain on a convex (n+2)-gon, implying a mixing time of Õ(n²). The previous state of the art for the mixing time of this chain due to Eppstein and Frishberg [Eppstein and Frishberg, 2023] was Õ(n³), while the best known lower bound on the mixing time due to Molloy, Reed and Steiger [Molloy et al., 1997] is Ω(n^{3/2}). Our relaxation time bound makes significant progress towards Aldous' [Aldous, 2003] conjectured bound of Θ(n^{3/2}) for the relaxation time.
We improve upon the analysis of [Eppstein and Frishberg, 2023] by further developing the framework of transport flows introduced in the work [Xiaoyu Chen et al., 2025] of Chen et al. In this light, our results can be seen as a more efficient way of using combinatorial decompositions to obtain functional inequalities for Markov chains. We hope our ideas will find other applications in the future.

Cite As Get BibTex

Vedat Levi Alev, Daniel Frishberg, Michail Sarantis, and Prasad Tetali. Faster Triangulation Mixing via Transport Flows. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.12

Author Details

Vedat Levi Alev
  • University of Haifa, Israel
Daniel Frishberg
  • California Polytechnic State University, San Luis Obispo, CA, USA
Michail Sarantis
  • Skyserv Handling Services, Athens, Greece
Prasad Tetali
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Alev, Vedat Levi: Supported by the ISF Grant No. 721/2024 of Uriya A. First.
  • Tetali, Prasad: Supported in part by the NSF DMS-2151283 grant and Alexander M. Knaster Professorship.

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