An Equilibrium Problem for the Limiting Eigenvalue Distribution of Banded Toeplitz Matrices

Abstract

Abstract. We study the limiting eigenvalue distribution of n×n banded Toeplitz matrices as n → ∞. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as n → ∞. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

“…Such functions appear as symbols of banded Toeplitz matrices [6], and we need certain results [7,13] that were derived in that context. Although we will not use Toeplitz matrices in this paper, we still refer to s as the symbol.…”

“…which are finite unions of analytic arcs and exceptional points, see [6,7]. A point z ∈ C for which the algebraic equation s(w) = z has a multiple solution is called a branch point.…”

“…Note that the right-hand side of the jth variational condition vanishes if z ∈ Γ j . In [7] there also appear constants ℓ 2 and ℓ 3 in (5.6) and (5.7). However, these constants vanish because Γ 2 and Γ 3 are unbounded and…”

“…, 1 along the diagonals for which it is known, see [6,7,11,17], that the eigenvalues accumulate on Γ 1 (ξ) as the size grows, with µ ξ 1 as limiting normalized eigenvalue counting measure. The distribution of the eigenvalues of (5.12) is then obtained by averaging of the measures µ ξ 1 as in (5.11).…”

“…By the continuity argument we obtain ( α(ξ), ∞) ⊂ Γ 1 (ξ) or ( α(ξ), ∞) ⊂ Γ 3 (ξ). Since Γ 1 (ξ) is a compact set, see [7], only the second inclusion holds.…”