Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: More Limiting Distributions for Eigenvalues of Wigner Matrices   The Tracy-Widom distributions are among the most famous laws in probability
theory, partly due to their connection with Wigner matrices. In particular, for
$A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$
symmetric with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. standard normal, the
fluctuations of its largest eigenvalue $\lambda_1(A)$ are asymptotically
described by a real-valued Tracy-Widom distribution $TW_1:$
$n^{2/3}(\lambda_1(A)-2) \Rightarrow TW_1.$ As it often happens, Gaussianity
can be relaxed, and this results holds when $\mathbb{E}[a_{11}]=0,
\mathbb{E}[a^2_{11}]=1,$ and the tail of $a_{11}$ decays sufficiently fast:
$\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=0,$ whereas when the law of
$a_{11}$ is regularly varying with index $\alpha \in (0,4),$
$c_a(n)n^{1/2-2/\alpha}\lambda_1(A)$ converges to a Fr\'echet distribution for
$c_a:(0,\infty) \to (0,\infty)$ slowly varying and depending solely on the law
of $a_{11}.$ This paper considers a family of edge cases, $\lim_{x \to
\infty}{x^4\mathbb{P}(|a_{11}|>x)}=c \in (0,\infty),$ and unveils a new type of
limiting behavior for $\lambda_1(A):$ a continuous function of a Fr\'echet
distribution in which $2,$ the almost sure limit of $\lambda_1(A)$ in the
light-tailed case, plays a pivotal role:
  $f(x)=\begin{cases}
  2, & 0<x<1 \newline
  x+\frac{1}{x}, & x \geq 1
  \end{cases}.$

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The categories of this paper are: q-fin
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The categories of this paper are: eess
29
The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: stat