Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nodal Lengths in Shrinking Domains for Random Eigenfunctions on
  $\mathbb{S}^2$   We investigate the asymptotic behavior of the nodal lines for random
spherical harmonics restricted to shrinking domains, in the 2-dimensional case:
i.e., the length of the zero set $\mathcal{Z}_{\ell,r_\ell} :=
\mathcal{Z}^{B_{r_{\ell}}}(T_\ell)=\text{len}(\{x \in \mathbb{S}^2 \cap
B_{r_\ell}: T_\ell(x)=0 \})$, where $B_{r_{\ell}}$ is the spherical cap of
radius $r_\ell$. We show that the variance of the nodal length is logarithmic
in the high energy limit; moreover, it is asymptotically fully equivalent, in
the $L^2$-sense, to the "local sample trispectrum", namely, the integral on the
ball of the fourth-order Hermite polynomial. This result extends and
generalizes some recent findings for the full spherical case. As a consequence
a Central Limit Theorem is established.

29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph