2015/11/23 by Miklós Kornyik, Kornyik, Miklós, György Michaletzky +1 · 2 citations
Mathematics · #15A52 #33C45 #60B20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1512.03724
openalex publication_date 2015/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present paper we give two alternate proofs of the well known theorem\nthat the empirical distribution of the appropriately normalized roots of the\nnth monic Hermite polynomial Hn converges weakly to the semicircle law,\nwhich is also the weak limit of the empirical distribution of appropriately\nnormalized eigenvalues of a Wigner matrix. In the first proof -- based on the\nrecursion satisfied by the Hermite polynomials -- we show that the generating\nfunction of the moments of roots of Hn is convergent and it satisfies a\nfixed point equation, which is also satisfied by c(z2), where c(z) is the\ngenerating function of the Catalan numbers Ck. In the second proof we\ncompute the leading and the second leading term of the kth moments (as a\npolynomial in n) of Hn and show that the first one coincides with\nCk/2, the (k/2) rm th Catalan number, where k is even and the\nsecond one is given by -(22k-1- binom2k-1k). We also mention the known\nresult that the expectation of the characteristic polynomial (pn) of a\nWigner random matrix is exactly the Hermite polynomial (Hn), i.e.\nEpn(x)=Hn(x), which suggest the presence of a deep connection between the\nHermite polynomials and Wigner matrices.\n