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On the Spectra and Pseudospectra of a Class of NonSelfAdjoint Random Matrices and Operators
, 2011
"... In this paper we develop and apply methods for the spectral analysis of nonselfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudoergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). ..."
Abstract

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In this paper we develop and apply methods for the spectral analysis of nonselfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudoergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semiinfinite and biinfinite matrix cases, for example showing that the numerical range and pnorm εpseudospectra (ε> 0, p ∈ [1, ∞]) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n × n matrices. We propose similar convergent approximations for the 2norm εpseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.