Abstract
Recently,M. and T. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states. We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of band matrices by Constantinescu. Other versions of this supersymmetry have been a crucial ingredient in the study of the localization–delocalization transition by theoretical physicists.
Highlights
Band operators and band matrices Random band operators are popular toy models of disordered systems in theoretical physics
For Gaussian random band matrices, the average |Gxy(E + iε)|2 corresponds, through Berezin integration and superbosonization or certain formal versions of the Hubbard–Stratonovich transformation [Zir06], to a high dimensional super-integral dominated by a complicated saddle manifold
Pointwise estimates The models (1.2) and their counterparts in higher dimension are especially convenient for supersymmetric analysis, since the dual supersymmetric model has nearest neighbour coupling
Summary
Band operators and band matrices Random band operators are popular toy models of disordered systems in theoretical physics. For Gaussian random band matrices including both (1.1) and (1.2), the density of states exists in both finite and infinite volume by a general argument of Wegner [Weg81]. For Gaussian random band matrices, the average |Gxy(E + iε)|2 corresponds, through Berezin integration and superbosonization or certain formal versions of the Hubbard–Stratonovich transformation [Zir06], to a high dimensional super-integral dominated by a complicated saddle manifold. Pointwise estimates The models (1.2) and their counterparts in higher dimension are especially convenient for supersymmetric analysis, since the dual supersymmetric model has nearest neighbour coupling (see Proposition 2.1; in classical statistical mechanics, the idea of duality between carefully chosen long-range models and nearest neighbour models goes back to the work of Mark Kac [Kac[59], Section 9]) This is why most of the pointwise results established to date pertain to this class of operators. With the exception of the Berezin integral representation (well explained elsewhere) and undergraduate analysis, our proof is self contained
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