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This is a Quantitative Study study.

October 12, 2025Open Access

Using dense graph limit theory to count cocycles of random simplicial complexes

Key Points

The dimension of H_1 converges to zero in probability for random 2-dimensional hypertrees, highlighting a major finding.
Using limit theory, this research presents an analogue to the large deviation principle for random cochains, establishing a novel approach.
This study applies techniques from dense graph limit theory to analyze the homology of random simplicial complexes, demonstrating significant relationships.
Results are corroborated for random 1-out 2-complexes, expanding the implications of the findings regarding random structures.

Abstract

We develop a limit theory for 1-cochains of complete graphs with coefficients from a finite abelian group. We prove an analogue of the large deviation principle of Chatterjee and Varadhan for random cochains. We use these new tools to prove results about the homology of random 2-dimensional simplicial complexes. More specifically, we prove that if Tₙ is a random 2-dimensional determinantal hypertree on n vertices and p is any prime, then \ H₁ (Tₙ, Fₚ) n²\ converges to zero in probability. The same result holds for random 1-out 2-complexes.

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Cite This Study

András Mészáros (Mon,) studied this question.

synapsesocial.com/papers/68ec1be02b8fa9b2b78ad109 https://doi.org/https://doi.org/10.48550/arxiv.2509.06559

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