Abstract
For each prime $p$, we study the eigenvalues of a 3-regular graph on roughly $p^2$ vertices constructed from the Markoff surface. We show they asymptotically follow the Kesten-McKay law, which also describes the eigenvalues of a random regular graph. The proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.
Highlights
The Kesten–McKay Law governs the eigenvalue distribution of a random d-regular graph in the limit of a growing number of vertices [Kes59, McK81]
For any fixed L or even up to a small multiple of log p, the path-count will approximately match what one would get in the process of computing a Kesten–McKay moment
Beyond the scale log p, the Markoff graph no longer resembles a tree in the same statistical sense that we have proved for smaller L
Summary
The Kesten–McKay Law governs the eigenvalue distribution of a random d-regular graph in the limit of a growing number of vertices [Kes, McK81]. Matthew DE COURCY-IRELAND & Michael MAGEE probability density function is (1.1) d ρd(λ) = 2π 4(d − 1) − λ2 d2 − λ2. This spectral density comes from the Plancherel measure on the infinite d-regular tree, and one might expect a similar eigenvalue distribution for non-random d-regular graphs provided they resemble their universal cover closely enough in the sense of having few short cycles. If an edge connects a vertex to itself, it must be counted just once in order for the graph to be 3-regular. The eigenvalues {λj} of the resulting graph can naturally be thought of as a measure on [−3, 3], namely (1.3)
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