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

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• 作者：Mikló ; s Kornyik ; ^{koma@cs.elte.hu" class="auth_mail" title="E-mail the corresponding author} ; Gyö ; rgy Michaletzky ^{michaletzky@caesar.elte.hu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A52 ; 60B20 ; 33C45
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：211
• 期：Complete
• 页码：29-41
• 全文大小：250 K

文摘

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$n$th monic Hermite polynomial H_n$H_n$ 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$H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by c(z²)$c\left(z^2\right)$, where c(z)$c\left(z\right)$ is the generating function of the Catalan numbers C_k$C_k$. In the second proof we compute the leading and the second leading term of the k$k$th moments (as a polynomial in n$n$) of H_n$H_n$ and show that the first one coincides with 122be29906c26cabbc0474" title="Click to view the MathML source">C_k/2$C_{k/2}$, the (k/2)$(k/2)$th Catalan number, where k$k$ is even and the second one is given by $-(2^{2k-1}-\left(\genfrac{}{}{00}{}{2k-1}{k}\right))$. We also mention the known result that the expectation of the characteristic polynomial (p_n$p_n$) of a Wigner random matrix is exactly the Hermite polynomial (H_n$H_n$), i.e. Ep_n(x)=H_n(x)$E{p}_n\left(x\right)=H_n\left(x\right)$, which suggests the presence of a deep connection between the Hermite polynomials and Wigner matrices.