# Smallest singular value distribution

Let $$G_\mathbb{R}\in\mathbb{R}^{n\times n}$$ and $$G_\mathbb{C}\in\mathbb{C}^{n\times n}$$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian entries of zero mean and variance $$\mathbb{E}\lvert(G_K)_{ij}\rvert^2=n^{-1}$$ for $$K=\mathbb{R},\mathbb{C}$$. Denote the singular values of the matrix $$X$$ by $$\sigma_1(X)\ge\dots\ge \sigma_n(X)$$. It follows from the work of Edelman that $$\begin{equation}\mathbb{P}\Big( \sigma_n(G_K)\le\frac{x}{n}\Big) = \begin{cases} 1-e^{-x^2/2-x}+o(1),&K=\mathbb{R}\\ 1-e^{-x^2},&K=\mathbb{C}. \end{cases}\tag{*}\label{sing val}\end{equation}$$ In particular, the smallest singular value is of order $$n^{-1}$$.

Now I am wondering about additive perturbations to $$G_K$$, say $$G_K+\lambda I$$, where $$\lambda$$ is some real (or even complex) parameter and $$I$$ is the identity matrix. In general the singular values of $$G_K$$ give little information about the singular values of $$G_K+\lambda I$$. The most one can hope for is bounds like $$\sigma_n(G_K+\lambda I)\ge \lvert\lambda\rvert-\sigma_1(G_K).$$ It is known that $$\sigma_1(G_K)$$ is Tracy-Widom distributed around $$2$$, so in particular, $$\sigma_n(G_k+\lambda I)$$ is bounded away from $$0$$ as long as $$\lvert \lambda\rvert>2$$.

Question: Is the analogue of $$\eqref{sing val}$$, i.e. the distribution of $$\sigma_n(G_K+\lambda I)$$ known for $$G_K+\lambda I$$? If exact formulae are not available, I would be interested in the average scaling of $$\sigma_n(G_K+\lambda I)$$. I guess there should be some phase transition of the type $$\mathbb E \sigma_n(G_K+\lambda I)\sim\begin{cases}n^{-1},&\lvert\lambda\rvertc.\end{cases}$$ I think $$c=1$$, but am unsure about the critical exponent.

• the scaling $1/n$ is always a lower bound, no matter what $\lambda$ is, e.g. by results of Sankar, Spielman and Teng cs.yale.edu/homes/spielman/Research/nopivotdas.pdf (Theorem 3.3). I am not sure about complementary upper bounds, especially in the critical regime – ofer zeitouni Apr 12 at 18:52