Sums of projections with random coefficients

Abstract

We study infinite sums \[ {\mathcal P}_{\varkappa}=\sum_{n=-\infty}^\infty \varkappa_n \langle\cdot, \psi_n\rangle\psi_n \] of rank-one projections in a Hilbert space, where $\{\psi_n\}_{n\in\mathbb Z}$ 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 $\{\langle\psi_n,\psi_m\rangle\}_{n,m\in\mathbb Z}$ 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 Schr\"odinger operators, we define the integrated density of states (IDS) measure $\nu_{{\mathcal P}_\varkappa}$ 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 $\psi_n$, we prove the Anderson-type localisation result: the spectrum of ${\mathcal P}_{\varkappa}$ is pure point almost surely.