On the heat kernel of a Cayley graph of PSL₂ℤ

2025/06/03 by Anders Karlsson, Karlsson, Anders, Kamila Kashaeva +1
Mathematics · Computer Science · #Graph theory and applications #Matrix Theory and Algorithms #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.2506.02340

Abstract

In this paper, we obtain an explicit formula for the heat kernel on the Cayley graph of the modular group PSL₂(Z), given by the presentation ⟨ a,b| a²=1, b³=1⟩. Our approach extends a method of Chung--Yau by observing that the Cayley graph strongly and regularly covers a weighted infinite line. We solve the spectral problem on this line to obtain an integral expression for its heat kernel, and then lift this to the Cayley graph using spectral transfer principles for strongly regular coverings. The explicit formula allows us to determine the Laplace spectrum, containing eigenvalues and continuous parts. As a by-product, we suggest a conjecture on the lower bound for the spectral gap of Cayley graphs of PSL₂\mathbbF_p with our generators, inspired by the analogy with Selberg's 1/4-conjecture. Numerical evidence to this conjecture is provided for small primes.

Citations

• Constructing heat kernels on infinite graphs
• On spectra and spectral measures of Schreier and Cayley graphs
• Quantum Chaos on random Cayley graphs of \rm SL₂[ℤ/pℤ]
• Growth in groups: ideas and perspectives
• Heat kernels on regular graphs and generalized Ihara zeta function formulas
• Explicit growth and expansion for SL₂
• Simplices and spectra of graphs
• Uniform expansion bounds for Cayley graphs of SL 2 (𝔽 p )

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