Sums of projections with random coefficients

Abstract

We study infinite sums \[ {\mathcal P}_{\varkappa}=\sum_{n=-\infty}^\infty \varkappa_n \langle\cdot, ψ_n\rangleψ_n \] of rank-one projections in a Hilbert space, where $\{\ddot{I}{ˆ}_n{\}}_{n\in \mathbb{Z}}$ are norm-one vectors, not necessarily orthogonal, and $\{{\varkappa }_n{\}}_{n\in \mathbb{Z}}$ are independent identically distributed positive random variables. Assuming that the Gram matrix $\{⟨\ddot{I}{ˆ}_n,\ddot{I}{ˆ}_m⟩{\}}_{n,m\in \mathbb{Z}}$ defines a bounded operator on ${\ell }^2(\mathbb{Z})$ and that its entries depend only on the difference $n-m$, we analyse ${\mathcal{P}}_{\varkappa }$ within the framework of spectral theory of ergodic operators. Inspired by the spectral theory of ergodic Schrödinger operators, we define the integrated density of states (IDS) measure $\hat{I}{½}_{{\mathcal{P}}_{\varkappa }}$ for ${\mathcal{P}}_{\varkappa }$ and establish results on its continuity and absolute continuity, including Wegner-type estimates and Lifshitz tail behaviour near the spectral edges. In the asymptotic regime of nearly-orthogonal $\ddot{I}{ˆ}_n$, we prove the Anderson-type localisation result: the spectrum of ${\mathcal{P}}_{\varkappa }$ is pure point almost surely.

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