Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random Matrices: The circular Law   Let $\a$ be a complex random variable with mean zero and bounded variance
$\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being
i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues
of $\frac{1}{\sigma \sqrt n}N_{n}$. Define the empirical spectral distribution
$\mu_{n}$ of $N_{n}$ by the formula $$ \mu_n(s,t) := \frac{1}{n} # \{k \leq n|
\Re(\lambda_k) \leq s; \Im(\lambda_k) \leq t \}.$$
  The Circular law conjecture asserts that $\mu_{n}$ converges to the uniform
distribution $\mu_\infty$ over the unit disk as $n$ tends to infinity.
  We prove this conjecture under the slightly stronger assumption that the
$(2+\eta)\th$-moment of $\a$ is bounded, for any $\eta >0$. Our method builds
and improves upon earlier work of Girko, Bai, G\"otze-Tikhomirov, and Pan-Zhou,
and also applies for sparse random matrices.
  The new key ingredient in the paper is a general result about the least
singular value of random matrices, which was obtained using tools and ideas
from additive combinatorics.

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The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg