2013/01/16 by Mireille Capitaine, Capitaine, Mireille · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Quantum optics and atomic interactions #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1301.3940
We consider large Information-Plus-Noise type matrices of the form\nMN=(\σ \(XN)/(\√(N))+AN)(\σ \(XN)/(\√(N))+AN)^*\nwhere XN is an n \× N (n\≤ N) matrix consisting of independent\nstandardized complex entries, AN is an n \× N nonrandom matrix and\n\σ>0. As N tends to infinity, if n/N \→ c\∈ ]0,1] and if\nthe empirical spectral measure of AN AN^* converges weakly to some\ncompactly supported probability distribution \ν \≠ \δ0, Dozier and\nSilverstein established that almost surely the empirical spectral measure of\nMN converges weakly towards a nonrandom distribution \μ\σ,\ν,c.\nBai and Silverstein proved, under certain assumptions on the model, that for\nsome closed interval in ]0;+\∞[ outside the support of\n\μ\σ,\ν,c satisfying some conditions involving AN, almost surely,\nno eigenvalues of MN will appear in this interval for all N large. In this\npaper, we carry on with the study of the support of the limiting spectral\nmeasure previously investigated by Dozier and Silverstein and later by Vallet,\nLoubaton and Mestre and Loubaton and P. Vallet, and we show that, under almost\nthe same assumptions as Bai and Silvertein, there is an exact separation\nphenomenon between the spectrum of MN and the spectrum of ANAN^*: to a\ngap in the spectrum of MN pointed out by Bai and Silverstein, it corresponds\na gap in the spectrum of ANAN^* which splits the spectrum of ANAN^*\nexactly as that of MN. We use the previous results to characterize the\noutliers of spiked Information-Plus-Noise type models.\n