Outliers of random perturbations of Toeplitz matrices with finite symbols
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Outliers of random perturbations of Toeplitz matrices with finite symbols Anirban Basak1
· Ofer Zeitouni2,3
Received: 3 June 2019 / Revised: 21 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract d1 Consider an N × N Toeplitz matrix TN with symbol a(λ) := =−d2 a λ , per−γ turbed by an additive noise matrix N E N , where the entries of E N are centered i.i.d. random variables of unit variance and γ > 1/2. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as N → ∞, to the law of a(U ), where U is distributed uniformly on S1 . In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N -independent) distance from a(S1 ). We prove that there are no outliers outside spec T (a), the spectrum of the limiting Toeplitz operator, with probability approaching one, as N → ∞. In contrast, in spec T (a)\a(S1 ) the process of outliers converges to the point process described by the zero set of certain random analytic functions. The limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d. having the same law as that of E N . The coefficients in the linear combination depend on the roots of the polynomial Pz,a (λ) := (a(λ) − z)λd2 and semi-standard Young Tableaux with shapes determined by the number of roots of Pz,a (λ) = 0 that are greater than one in moduli. Mathematics Subject Classification Primary 60B20; Secondary 15A18 · 47A55 · 47B80
B
Anirban Basak [email protected]
1
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
2
Department of Mathematics, Weizmann Institute of Science, POB 26, 76100 Rehovot, Israel
3
Courant Institute, New York University, 251 Mercer St, New York, NY 10012, USA
123
A. Basak, O. Zeitouni
1 Introduction Let a : C → C be a Laurent polynomial. That is, d1
a(λ) :=
a λ , λ ∈ C,
(1.1)
=−d2 1 , so that for some d1 , d2 ∈ N and some sequence of complex numbers {a }d=−d 2 N N 1 d1 > 0 . Define T (a) : C → C to be the Toeplitz operator with symbol a, that is the operator given by
(T (a)x)i :=
d1
a x+i ,
where x := (x1 , x2 , . . .) ∈ CN ,
for i ∈ N,
=−d2
and we set xi = 0 for non-positive integer values of i. For N ∈ N, we denote by TN (a) the natural N -dimensional truncation of the infinite dimensional Toeplitz operator T (a). As a matrix of dimension N × N , we have (for N > max(d1 , d2 )) that ⎡
a0
⎢ ⎢a−1 ⎢ ⎢ ⎢a−2 TN := TN (a) := ⎢ ⎢ . ⎢ .. ⎢ ⎢ . ⎣ .. 0
a−1 .. .
..
.
··· .. . .. . .. .
···
. ···
a−1 a−2
a1
a2
a0
a1 .. .
..
···
a0 a−1
..
.
a1
⎤ 0 .. ⎥ .⎥ ⎥ .. ⎥ .⎥ ⎥. ⎥ a2 ⎥ ⎥ ⎥ a1 ⎦ a0
In general, TN is not a normal matrix, and thus its spectrum can be sensitive to small perturbations. In this paper, we will be interested in the spectrum of M N := TN + N , where N is a “vanishing” random perturbation, and especially in outliers, i.e. eigenvalues that are a
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