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Circular law for random discrete matrices of given row sum

Let $M_n$ be a random matrix of size $n\times n$ 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 $$μ_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(λ_k)\le s; \Im(λ_k)\le t\}.$$ 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 $\frac{1}{σ\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution $μ_\cir$ 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 $|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of $M_n$.