Kesten–McKay law for the Markoff surface mod p
[Loi de Kesten–McKay pour la surface de Markoff modulo p]
de Courcy-Ireland, Matthew1 ; Magee, Michael2
1 Department of Mathematics, Princeton University, Princeton, NJ, 08544, (USA)
2 Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, (UK)
Annales Henri Lebesgue, Tome 4 (2021), pp. 227-250.

Pour chaque nombre premier p, on décrit les valeurs propres d’un graphe 3-régulier ayant environ p 2 sommets construit à partir de la surface de Markoff. On montre qu’elles suivent approximativement la loi de Kesten–McKay, qui décrit également les valeurs propres d’un graphe aléatoire régulier. On utilise la méthode des moments et l’action de GL 2 (ℤ) sur la surface de Markoff.

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.

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DOI : 10.5802/ahl.71
Classification : 11D25, 05C50, 11F72, 37P25
Mots clés : Markoff surface, Kesten–McKay law, cubic surfaces, graphs and groups
de Courcy-Ireland, Matthew 1 ; Magee, Michael 2

1 Department of Mathematics, Princeton University, Princeton, NJ, 08544, (USA)
2 Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, (UK) 
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     title = {Kesten{\textendash}McKay law for the {Markoff} surface mod $p$},
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de Courcy-Ireland, Matthew; Magee, Michael. Kesten–McKay law for the Markoff surface mod $p$. Annales Henri Lebesgue, Tome 4 (2021), pp. 227-250. doi : 10.5802/ahl.71. http://www.numdam.org/articles/10.5802/ahl.71/

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