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On the least singular value of random symmetric matrices

   
@article{EJP2165,
	author = {Hoi Nguyen},
	title = {On the least singular value of random symmetric matrices},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {Random symmetric matrices, least singular values},
	abstract = {Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a {\it random symmetric matrix} whose upper diagonal entries $x_{ij}, 1\le i\le j,$ are iid copies of a random variable $\xi$. Under a very general assumption on $\xi$, we show that for any $B>0$ there exists $A>0$ such that $\mathbb{P}(\sigma_n(M_n)\le n^{-A})\le n^{-B}$.},
	pages = {no. 53, 1-19},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-2165},    
        url = {http://ejp.ejpecp.org/article/view/2165}}