Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Wigner matrices, the moments of roots of Hermite polynomials and the
  semicircle law   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.

31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga