The Least Singular Value of the General Deformed Ginibre Ensemble

Abstract

We study the least singular value of the $n\times n$ matrix $H-z$ with $H=A_0+H_0$, where $H_0$ is drawn from the complex Ginibre ensemble of matrices with iid Gaussian entries, and $A_0$ is some general $n\times n$ matrix with complex entries (it can be random and in this case it is independent of $H_0$). Assuming some rather general assumptions on $A_0$, we prove an optimal tail estimate on the least singular value in the regime where z is around the spectral edge of H thus generalize the recent result of Cipolloni et al. (Probab Math Phys 1(1):101–146, 2020) to the case $A_0\ne 0$. The result improves the classical bound by Sankar et al. (SIAM J Matrix Anal Appl 28:446–476, 2006).

Appendix

1.1 Proof of Proposition 3.1

Let us introduce the averaging (see (3.26))

$$\begin{aligned}&\langle f\rangle =\frac{4}{\pi ^2}\Re \Big [e^{i\pi /4}\int _0^\infty f({\tilde{u}},\theta ) p({\tilde{u}},{\tilde{\tau }})e^{{\tilde{F}}_1({\tilde{u}})}d{\tilde{u}} \int _{0}^\infty e^{{\tilde{F}}_2({\tilde{\tau }})}d{\tilde{\tau }}\int _{-\infty }^\infty e^{-{\tilde{\varepsilon }}{\tilde{v}}^2}d{\tilde{v}}\\&\quad \int _{0}^{2\pi }e^{-2{\tilde{\varepsilon }}({\tilde{u}}_*+{\tilde{u}}) (\cos \theta +1) }d\theta \Big ]\\&{\tilde{F}}_1({\tilde{u}})=-c_2{\tilde{u}}^4/2-a_2{\tilde{u}}^2-a_3{\tilde{u}}^3\\&{\tilde{F}}_2({\tilde{\tau }})= -c_2{\tilde{\tau }}^4/2-ia_2{\tilde{\tau }}^2-ia_3e^{ i\pi /4}{\tilde{\tau }}^3\\&p({\tilde{u}},{\tilde{\tau }}):=({\tilde{u}}+{\tilde{u}}_*) ({\tilde{u}}_*-ie^{ i\pi /4}{\tilde{\tau }})\frac{{\tilde{\varphi }}({\tilde{u}},e^{ i\pi /4}{\tilde{\tau }})}{({\tilde{v}}^2+4{\tilde{u}}_*-4ie^{ i\pi /4}{\tilde{\tau }})^{1/2}}, \end{aligned}$$

where ${\tilde{\varphi }}$ and $a_{2}$ and $a_3$ are defined in (3.24). Then by construction

$$\begin{aligned} 1=&{\mathcal {Z}}(\varepsilon ,\varepsilon )=\left\langle 1\right\rangle +O(n^{-1/2}). \end{aligned}$$

(5.1)

Moreover, evidently,

$$\begin{aligned} I({\tilde{\delta }},{\tilde{\varepsilon }})=-{\tilde{\varepsilon }}^{-1}\langle ({\tilde{u}}+{\tilde{u}}_*)\cos \theta \rangle =-\frac{{\tilde{u}}_*}{{\tilde{\varepsilon }}}\langle \cos \theta \rangle -\frac{1}{{\tilde{\varepsilon }}}\langle {\tilde{u}}\cos \theta \rangle . \end{aligned}$$

(5.2)

Equation (3.25), as $\varepsilon \rightarrow \infty$, yields:

$$\begin{aligned} {\tilde{u}}_*({\tilde{\delta }},{\tilde{\varepsilon }})=&({\tilde{\varepsilon }}/c_2)^{1/3}(1+o(1))\rightarrow \infty , \quad a_2=3c_2{\tilde{u}}_*^2(1+o(1))\rightarrow \infty ,\quad a_3=2c_2{\tilde{u}}_*\rightarrow \infty . \end{aligned}$$

Since $a_2\rightarrow \infty$ and

$$\begin{aligned} \max {\tilde{F}}_1({\tilde{u}})={\tilde{F}}_1(0)=0,\quad \max \{-(\cos \theta +1)\}=0,\quad \max \{-{\tilde{v}}^2\}=0,\end{aligned}$$

one can use a standard saddle point method for integration with respect to ${\tilde{u}}$, $\theta$ and ${\tilde{v}}$. To integrate with respect to ${\tilde{\tau }}$ we move the contour of integration from ${\mathbb {R}}_+$ to $e^{-i\pi /8}{\mathbb {R}}_+$. Then

$$\begin{aligned}\max \Re {\tilde{F}}_2(e^{-i\pi /8}{\tilde{\tau }} )=\max \Re (-a_2e^{i\pi /4}{\tilde{\tau }}^2- a_3e^{3i\pi /8}{\tilde{\tau }}^3)={\tilde{F}}_2(0)=0,\end{aligned}$$

and we can apply the saddle point method for the integral with respect to ${\tilde{\tau }}$ also. We restrict the integration domain to

$$\begin{aligned} |{\tilde{u}}|\le {\tilde{u}}_*^{-1}\log {\tilde{u}}_*,\quad |{\tilde{\tau }}|\le {\tilde{u}}_*^{-1}\log {\tilde{u}}_*,\quad |v|\le {\tilde{\varepsilon }}^{-1/2}\log {\tilde{\varepsilon }} \quad |\theta -\pi |\le ({\tilde{\varepsilon }}{\tilde{u}}_*)^{-1/2}\log ({\tilde{\varepsilon }}{\tilde{u}}_*) \end{aligned}$$

and change the variables

$$\begin{aligned} {\tilde{u}}=&u'/{\tilde{u}}_*,\quad {\tilde{\tau }}= e^{-i\pi /8}\tau '/{\tilde{u}}_*,\quad {\tilde{v}}=v'/{\tilde{\varepsilon }}^{1/2},\quad \theta =-\pi +\theta '({\tilde{\varepsilon }}{\tilde{u}}_*)^{-1/2} \end{aligned}$$

Then

$$\begin{aligned}&{\tilde{F}}_1({\tilde{u}})\rightarrow -3c_2(u')^2+O({\tilde{u}}_*^{-1});\nonumber \\&{\tilde{F}}_1({\tilde{\tau }})\rightarrow -3c_2e^{i\pi /4}(\tau ')^2+O({\tilde{u}}_*^{-1});\nonumber \\&\quad -2{\tilde{\varepsilon }}{\tilde{u}}_*(\cos \theta +1)\rightarrow -(\theta ')^2+O(({\tilde{\varepsilon }}{\tilde{u}}_*)^{-1/2});\nonumber \\&p({\tilde{u}},{\tilde{\tau }})\rightarrow 3({\tilde{u}}_*)^6c_2^2/2({\tilde{u}}_*)^{1/2}(1+O({\tilde{u}}_*^{-1/2})), \end{aligned}$$

(5.3)

and substituting this to (5.2) we get

$$\begin{aligned} I({\tilde{\delta }},{\tilde{\varepsilon }})=&\frac{{\tilde{u}}_*}{{\tilde{\varepsilon }}}\langle 1\rangle _0(1+O({\tilde{u}}_*^{-1/2})) +O(\frac{1}{{\tilde{\varepsilon }}{\tilde{u}}_*}) \end{aligned}$$

(5.4)

where we set

$$\begin{aligned} \langle f\rangle _0=&\frac{6c_2^2({\tilde{u}}_*)^3}{\pi ^2{\tilde{\varepsilon }}} \Re \Big [e^{i\pi /8}\int du'd\tau 'd\theta 'dv' f(u',\theta ')e^{-3c_2(u')^2-3c_2e^{i\pi /4}(\tau ')^2-(\theta ')^2-(v')^2}\Big ] \end{aligned}$$

By (5.1) and (5.3) we have

$$\begin{aligned} \langle 1\rangle _0=1+O({\tilde{u}}_*^{-1/2}) \end{aligned}$$

and combining the above relation with (5.4), we obtain (3.27). $\square$