Asymptotic eigenvalue analysis for a sparse random matrix

We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix: $H=\{h_{ij}\}_{N_r\times N_t}$,

where each entry $h_{ij}$ is with a probability $p$ to obey the Gaussian distribution $N(0,σ^2)$, and with a probability $1-p$ to be zero.

Then we have following questions

a)Can we use a specific distribution (pdf) to describe the entries of this $N_r \times N_t$ matrix as $h_{ij}\sim (1-p)\delta(0)+p\frac{1}{\sigma\sqrt{2\pi}}e^{-j\frac{x^2}{2\sigma^2}}\delta(x)$?

b)When $N_r,N_t\rightarrow \infty$, and $N_r/N_t\rightarrow \beta$ (determined value), does the eigenvalues’ PDF go to a deterministic distribution? If the answer is Yes, what is the explicit expression of this deterministic distribution?

For question b), one potentially useful tool maybe the Mar˘cenko-Pastur law in the random matrix theory, but I do not know if it can be used indeed.

• The matrices aren't square. Do you mean singular values instead of eigenvalues? – Lior Silberman Sep 29 '14 at 13:06
• Yes, for a general case, singular values should be considered, and the Marchenko-Pastur law may can be used. But from my comparison between the asymptotic results and the numerical results, there is a mismatch, I cannot explain the reason. – Wenlong Cai Sep 30 '14 at 0:12

1 Answer

I don't really understand your question (a) - but if you look up "mixtures", it will be answered. As for question (b), the magic words are "Girko's circular law", and the magic reference is the Wikipedia.

• For question(a), I think the entries follow the distribution something like the Gaussian-Bernoulli distribution in the compress sensing applications. For question (b), I do not quite sure if the Mar˘cenko-Pastur law in the random matrix theory can be used indeed. – Wenlong Cai Sep 29 '14 at 3:19
• Marchenko-Pastur is about SINGULAR values, and has nothing to do with this question. I have answered your question. – Igor Rivin Sep 29 '14 at 3:40
• Thanks for your help, I will check your reference. I have another question about the circular law, does the circular law suit for all the distributions, or there is a limitation on the distribution of the entries? – Wenlong Cai Sep 29 '14 at 4:11
• @WenlongCai that is discussed in the wikipedia article, with references (in particular, to the Tao-Vu paper) – Igor Rivin Sep 29 '14 at 4:16
• I run a simulation on this kind of sparse random matrix, and compared the eigenvalues' PDF simulation results with the asymptotic results from the random matrix book, there is mismatch between these results. I will double check the simulation program. – Wenlong Cai Sep 29 '14 at 6:31