Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The arc length of a random lemniscate   A polynomial lemniscate is a curve in the complex plane defined by $\{z \in
\mathbb{C}:|p(z)|=t\}$. Erd\"os, Herzog, and Piranian posed the extremal
problem of determining the maximum length of a lemniscate $\Lambda=\{ z \in
\mathbb{C}:|p(z)|=1\}$ when $p$ is a monic polynomial of degree $n$. In this
paper, we study the length and topology of a random lemniscate whose defining
polynomial has independent Gaussian coefficients. In the special case of the
Kac ensemble we show that the length approaches a nonzero constant as $n
\rightarrow \infty$. We also show that the average number of connected
components is asymptotically $n$, and we observe a positive probability
(independent of $n$) of a giant component occurring.

20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc