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- W4301508066 abstract "In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic 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 $pm 1$'s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and $p$-norm $eps$-pseudospectra ($eps>0$, $pin [1,infty]$) 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 $Sigma$. We also propose a sequence of inclusion sets for $Sigma$ which we show is convergent to $Sigma$, with the $n$th element of the sequence computable by calculating smallest singular values of (large numbers of) $ntimes n$ matrices. We propose similar convergent approximations for the 2-norm $eps$-pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below." @default.
- W4301508066 created "2022-10-05" @default.
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- W4301508066 date "2011-07-01" @default.
- W4301508066 modified "2023-10-06" @default.
- W4301508066 title "On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators" @default.
- W4301508066 doi "https://doi.org/10.48550/arxiv.1107.0177" @default.
- W4301508066 hasPublicationYear "2011" @default.
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