Limit of Infinite Band Width for Product of Two Random Matrices
We study eigenvalue distribution of N*N random matrices HN having the form of a product
ANANT restricted to the band of the width 2R. We consider the case
when entries of random matrices AN are independent identically distributed Gaussian random variables.
We prove that the normalized eigenvalue counting function of HN converges in probability
as N,R tend to infinity to a nonrandom distribution s(l).
We derive equation for the Stieltjes transform of s(l),
which depends on the ratio b=lim R/N whereas N,R tend to infinity, and show that
s(l) coincides with the Wigner semicircle distribution
when b=0. The latter means that transition to the narrow band eliminates dependences between entries
ANANT.
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