Regularization of non-normal matrices by Gaussian noise - the banded\n Toeplitz and twisted Toeplitz cases

2017/11/30 by Anirban Basak, Basak, Anirban, Elliot Paquette +3 · 1 citation
Mathematics · Computer Science · #Random Matrices and Applications #Matrix Theory and Algorithms #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1712.00042

Abstract

We consider the spectrum of additive, polynomially vanishing random\nperturbations of deterministic matrices, as follows. Let M_N be a\ndeterministic N\× N matrix, and let G_N be a complex Ginibre matrix. We\nconsider the matrix \M_N=M_N+N^-\γG_N, where \γ>1/2.\nWith L_N the empirical measure of eigenvalues of \M_N, we provide\na general deterministic equivalence theorem that ties L_N to the singular\nvalues of z-M_N, with z\∈ \ℂ. We then compute the limit of L_N\nwhen M_N is an upper triangular Toeplitz matrix of finite symbol: if\nM_N=\∑_i=0^ mathfrakd a_i Jⁱ where mathfrakd is fixed,\na_i\∈\C are deterministic scalars and J is the nilpotent matrix\nJ(i,j)= bf 1_j=i+1, then L_N converges, as N\→\∞, to the law of\n\∑_i=0^ mathfrakd a_i Uⁱ where U is a uniform random variable on\nthe unit circle in the complex plane. We also consider the case of slowly\nvarying diagonals (twisted Toeplitz matrices), and, when mathfrakd=1, also\nof i.i.d.~entries on the diagonals in M_N.\n

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