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In this article, we shall start with a closed walk on a regular tree ofdegree $d$. These walks are described by the Kesten-McKay law which arises asthe asymptotic distribution of a random $d$-regular graph on $n$ vertices. Wewill show that the moments of the Kesten-McKay law are given by countingstandard Young tableaux with at most 2 rows, and how some properties of thewalk make sense even when $d$ is not an integer. We will use free probabilityto instruct us how to build an explicit model in random matrix theory.

Freely Independent Coin Tosses, Standard Young Tableaux, and the Kesten--McKay Law