Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law

TL;DR

This paper provides two alternative proofs that the roots of Hermite polynomials, when normalized, follow the semicircle law, highlighting their connection to Wigner matrices and moments related to Catalan numbers.

Contribution

It introduces two new proofs of the convergence of Hermite polynomial roots to the semicircle law, using recursion and moment calculations, revealing deep links to Wigner matrices.

Findings

Roots of Hermite polynomials follow the semicircle law.

Moments of roots relate to Catalan numbers.

Expectation of Wigner matrix characteristic polynomial equals Hermite polynomial.

Abstract

In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof -- based on the recursion satisfied by the Hermite polynomials -- we show that the generating function of the moments of roots of $H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by $c(z^2)$, where $c(z)$ is the generating function of the Catalan numbers $C_k$. In the second proof we compute the leading and the second leading term of the $k^{th}$ moments (as a polynomial in $n$) of $H_n$ and show that the first one coincides with $C_{k/2}$, the $(k/2)^{\rm th}$ Catalan number, where $k$ is even and the second one is given by $-(2^{2k-1}-\binom{2k-1}{k})$. We also mention the known result that the expectation of the characteristic polynomial ($p_n$) of a Wigner random matrix is exactly the Hermite polynomial ($H_n$), i.e. $Ep_n(x)=H_n(x)$, which suggest the presence of a deep connection between the Hermite polynomials and Wigner matrices.