2012/03/27 by Nguyen, Hoi H., Vu, Van
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1203.5941
Let Mn be a random matrix of size n× n and let λ1,...,λn be the eigenvalues of Mn. The empirical spectral distribution μMn of Mn is defined as μMn(s,t)=(1)/(n)# \k≤ n, \Re(λk)≤ s; \Im(λk)≤ t\. The circular law theorem in random matrix theory asserts that if the entries of Mn are i.i.d. copies of a random variable with mean zero and variance σ2, then the empirical spectral distribution of the normalized matrix (1)/(σ√(n))Mn of Mn 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 Mn, 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 Mn.