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

2025/09/08 by Mészáros, András · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.06559

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 T_n is a random 2-dimensional determinantal hypertree on n vertices and p is any prime, then \fracdim H₁(T_n,\mathbbF_p)n² converges to zero in probability. The same result holds for random 1-out 2-complexes.

Citations

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Cited by

• Local limits of determinantal processes

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• Bounds on the mod 2 homology of random 2-dimensional determinantal hypertrees
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• Vanishing of cohomology groups of random simplicial complexes