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Register a new userSpectrum and pseudospectrum for quadratic polynomials in Ginibre matrices
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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.
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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.
In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $operatorname{GL}_n(mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.
Let $O(2n+ell)$ be the group of orthogonal matrices of size $left(2n+ellright)times left(2n+ellright)$ equipped with the probability distribution given by normalized Haar measure. We study the probability begin{equation*} p_{2n}^{left(ellright)} = mathbb{P}left[M_{2n} , mbox{has no real eigenvalues}right], end{equation*} where $M_{2n}$ is the $2ntimes 2n$ left top minor of a $(2n+ell)times(2n+ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $ntimes n$ that depends on the truncation parameter $ell$. For $ell=1$ the matrix coincides with the Hilbert matrix and we prove begin{equation*} p_{2n}^{left(1right)} sim n^{-3/8}, mbox{ when }n to infty. end{equation*} We also discuss connections of the above to the persistence probability for random Kac polynomials.
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescus sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.
We propose a boundary regularity condition for the $M_n(mathbb{C})$-valued subordination functions in free probability to prove the local limit theorem and delocalization of eigenvectors for polynomials in two random matrices. We prove this through estimating the pair of $M_n(mathbb{C})$-valued approximate subordination functions for the sum of two $M_n(mathbb{C})$-valued random matrices $gamma_1otimes C_N+gamma_2otimes U_N^*D_NU_N$, where $C_N$, $D_N$ are deterministic diagonal matrices, and $U_N$ is Haar unitary.
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