Circular law for random discrete matrices of given row sum
Abstract
Let
M
n
be a random matrix of size
n×n
and let
λ
1
,...,
λ
n
be the eigenvalues of
M
n
. The empirical spectral distribution
μ
M
n
of
M
n
is defined as
\mu_{M_n}(s,t)=\frac{1}{n}#
\{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}.
The circular law theorem in random matrix theory asserts that if the entries of
M
n
are i.i.d. copies of a random variable with mean zero and variance
σ
2
, then the empirical spectral distribution of the normalized matrix
1
σ
n
√
M
n
of
M
n
converges almost surely to the uniform distribution $\mu_\cir$ over the unit disk as
n
tends to infinity.
In this paper we show that the empirical spectral distribution of the normalized matrix of
M
n
, a random matrix whose rows are independent random
(−1,1)
vectors of given row-sum
s
with some fixed integer
s
satisfying
|s|≤(1−o(1))n
, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of
M
n
.