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

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.