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@article{ECP3027,
	author = {Florent Benaych-Georges and Sandrine Péché},
	title = {Largest eigenvalues and eigenvectors of band or sparse random matrices},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {19},
	year = {2014},
	keywords = {random matrices ; band matrices ; largest eigenvalues ; localization},
	abstract = {
In this text, we consider an $N$ by $N$ random matrix $X$ such that all but $o(N)$ rows of $X$ have $W$ non identically zero entries, the other rows having less than $W$ entries  (such as, for example, standard or cyclic band matrices). We always suppose that  $1 \ll W \ll N$. We first prove that if the entries are independent, centered,  have variance  one, satisfy a certain tail upper-bound condition and $W \gg (\log N)^{6(1+\alpha)}$, where $\alpha$ is a positive parameter depending on the distribution of the entries, then the  largest eigenvalue of $X/\sqrt{W}$ converges to the upper bound of its limit spectral distribution, that is $2$, as for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin where less hypotheses were made on $W$, but more hypotheses were made about the law of the entries and the structure of the matrix. Then, under the same hypotheses, we prove a delocalization result for the eigenvectors of $X$, precisely  that most of them cannot be essentially localized on less than $W/\log(N)$ entries. This lower bound on the localization length has to be compared to the recent result by Steinerberger, which states that the localization length in the edge is $\ll W^{7/5}$ or there is strong interaction between two eigenvectors in an interval oflength $W^{7/5}$.
},
	pages = {no. 4, 1-9},
	issn = {1083-589X},
	doi = {10.1214/ECP.v19-3027},    
        url = {http://ecp.ejpecp.org/article/view/3027}}