Sums of projections with random coefficients
Leonid Pastur, Alexander Pushnitski
TL;DR
This work analyzes infinite sums of random rank-one projections ${\mathcal P}_{\varkappa}=\sum_{n} \varkappa_n \langle\cdot,\psi_n\rangle\psi_n$ in a Hilbert space, with a translation-invariant Gram matrix $G$ encoded by the symbol $\varphi$ through the Laurent operator $L_{\varphi}$. By exploiting ergodicity, they prove the spectrum is almost surely deterministic and define the integrated density of states $\nu_{\mathcal{P}_{\varkappa}}$, establishing a Wegner-type bound and Lifshitz tails for the IDS; they also derive explicit moment formulas linking $\nu_{\mathcal{P}_{\varkappa}}$ to $\mathbf E[\varkappa_0]$, $\mathbf V[\varkappa_0]$, and the Fourier coefficients of $\varphi$. The paper provides concrete realizations in $\ell^2(\mathbb{Z})$ and $L^2(\mathbb{T})$, relates the spectrum to multiplicative perturbations of Laurent operators, and proves Anderson localisation in the nearly-orthogonal regime via fractional moments and Simon–Wolff techniques. These results extend ergodic operator theory to a nonlocal random-projection setting and connect the spectral properties to a rich structure of the Gram symbol and Beurling–Mallivin theory for possible atoms in the IDS. Overall, the work offers a rigorous framework for understanding randomness in sums of projections, with implications for nonlocal random operators and ergodic spectral theory.
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 $\{ψ_n\}_{n\in\mathbb Z}$ 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 $\{\langleψ_n,ψ_m\rangle\}_{n,m\in\mathbb Z}$ 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 Schrödinger operators, we define the integrated density of states (IDS) measure $ν_{{\mathcal P}_\varkappa}$ 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 $ψ_n$, we prove the Anderson-type localisation result: the spectrum of ${\mathcal P}_{\varkappa}$ is pure point almost surely.