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Register a new userOverlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations
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Added by
Nikhil Srivastava
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Let $G_n$ be an $n times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n times n$ matrix with $Vert A Vert le 1$, and let $gamma in (0,1)$. We show that with probability $0.99$, $A + gamma G_n$ has all of its eigenvalue condition numbers bounded by $Oleft(n^{5/2}/gamma^{3/2}right)$ and eigenvector condition number bounded by $Oleft(n^3 /gamma^{3/2}right)$. Furthermore, we show that for any $s > 0$, the probability that $A + gamma G_n$ has two eigenvalues within distance at most $s$ of each other is $Oleft(n^4 s^{1/3}/gamma^{5/2}right).$ In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work [Banks et al. 2019] which proved an eigenvector condition number bound of $Oleft(n^{3/2} / gammaright)$ for the simpler case of {em complex} i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts $z-(A+gamma G_n)$ which recover the tail behavior of the complex Ginibre ensemble when $Im z eq 0$. This yields sharp control on the area of the pseudospectrum $Lambda_epsilon(A+gamma G_n)$ in terms of the pseudospectral parameter $epsilon>0$, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.
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For a fixed quadratic polynomial $mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $Ntimes N$ complex Ginibre matrices $X_1^N,dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =mathfrak{p}(X_1^N,dots, X_n^N)$ to the Brown measure of $mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.
Let $sqrt{N}+lambda_{max}$ be the largest real eigenvalue of a random $Ntimes N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix). We study the large deviations behaviour of the limiting $Nrightarrow infty$ distribution $P[lambda_{max}<t]$ of the shifted maximal real eigenvalue $lambda_{max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, [ P[lambda_{max}<t]=1-frac{1}{4}mbox{erfc}(t)+Oleft(e^{-2t^2}right). ] This is a rigorous confirmation of the corresponding result of Forrester and Nagao. We also prove that the left tail is exponential: for $t<0$, [ P[lambda_{max}<t]= e^{frac{1}{2sqrt{2pi}}zetaleft(frac{3}{2}right)t+O(1)}, ] where $zeta$ is the Riemann zeta-function. Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABMs) with the step initial condition. Therefore, the tail behaviour of the distribution of $X_s^{(max)}$ - the position of the rightmost annihilating particle at fixed time $s>0$ - can be read off from the corresponding answers for $lambda_{max}$ using $X_s^{(max)}stackrel{D}{=} sqrt{4s}lambda_{max}$.
We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable $(1/2)$ subordinator$.$ The required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001).
We establish a quantitative criterion for an operator defined on a Galton-Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative variance below a threshold. As a byproduct, we prove that the adjacency operator of a supercritical Poisson Galton-Watson tree has a non-trivial absolutely continuous part if the average degree is large enough. We also prove that its Karp and Sipser core has purely absolutely spectrum on an interval if the average degree is large enough. We finally illustrate our criterion on the Anderson model on a d-regular infinite tree with d $ge$ 3 and give a quantitative version of Kleins Theorem on the existence of absolutely continuous spectrum at disorder smaller that C $sqrt$ d for some absolute constant C.
In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of $k$-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on $kgeq2$ complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.
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