In this manuscript we give an extension of the classic Salem–Zygmund inequality for locally sub-Gaussian random variables. As an application, the concentration of the roots of a Kac polynomial is studied, which is the main contribution of this manuscript. More precisely, we assume the existence of the moment generating function for the iid random coefficients for the Kac polynomial and prove that there exists an annulus of width $$\begin{aligned} \text {O}( n^{-2}(\log n)^{-1/2-\gamma }), \quad \gamma >1/2\end{aligned}$$ O ( n - 2 ( log n ) - 1 / 2 - γ ) , γ > 1 / 2 around the unit circle that does not contain roots with high probability. As an another application, we show that the smallest singular value of a random circulant matrix is at least $$n^{-\rho }$$ n - ρ , $$\rho \in (0,1/4)$$ ρ ∈ ( 0 , 1 / 4 ) with probability $$1-\text {O}( n^{-2\rho })$$ 1 - O ( n - 2 ρ ) .
In this paper, we study how the roots of the Kac polynomials $$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$ W n ( z ) = ∑ k = 0 n - 1 ξ k z k concentrate around the unit circle when the coefficients of $$W_n$$ W n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as $$n\rightarrow \infty $$ n → ∞ if and only if $${\mathbb {E}}[\log ( 1+ |\xi _0|)]<\infty $$ E [ log ( 1 + | ξ 0 | ) ] < ∞ . Under the condition $${\mathbb {E}}[\xi ^2_0]<\infty $$ E [ ξ 0 2 ] < ∞ , we show that there exists an annulus of width $${\text {O}}(n^{-2}(\log n)^{-3})$$ O ( n - 2 ( log n ) - 3 ) around the unit circle which is free of roots with probability $$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$ 1 - O ( ( log n ) - 1 / 2 ) . The proof relies on small ball probability inequalities and the least common denominator used in [17].
We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity, which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.
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