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Register a new userEigenvalues of the non-backtracking operator detached from the bulk
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Added by
Yizhe Zhu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{mathrm{in}}, p_{mathrm{out}} = omega(log n)/n$. In this regime we answer a question posed in DallAmico and al. (2019) regarding the existence of a real eigenvalue `inside the bulk, close to the location $frac{p_{mathrm{in}}+ p_{mathrm{out}}}{p_{mathrm{in}}- p_{mathrm{out}}}$. We also introduce a variant of the Bauer-Fike theorem well suited for perturbations of quadratic eigenvalue problems, and which could be of independent interest.
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For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).
We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $pm infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.
In this paper, we present a method for the computation of the low-lying real eigenvalues of the Wilson-Dirac operator based on the Arnoldi algorithm. These eigenvalues contain information about several observables. We used them to calculate the sign of the fermion determinant in one-flavor QCD and the sign of the Pfaffian in N=1 super Yang-Mills theory. The method is based on polynomial transformations of the Wilson-Dirac operator, leading to considerable improvements of the computation of eigenvalues. We introduce an iterative procedure for the construction of the polynomials and demonstrate the improvement in the efficiency of the computation. In general, the method can be applied to operators with a symmetric and bounded eigenspectrum.
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