Markov Inequality as a Tool for Linear-Scaling Estimation of Local Observables

TL;DR

This paper presents a linear-scaling stochastic method for accurately estimating local spectral properties in disordered lattice models, using Markov inequality to bound sampling errors and extend to non-diagonal observables.

Contribution

The authors introduce a novel stochastic approach employing Markov inequality for rigorous error bounds, enabling efficient and accurate local observable estimation in large disordered systems.

Findings

Successfully computed local density of states and currents in disordered 2D models.

Demonstrated the method's effectiveness in handling strong spatial fluctuations.

Enabled simulations of large-scale mesoscopic phenomena.

Abstract

We introduce a linear-scaling stochastic method to compute real-space maps of any positive local spectral operator in a tight-binding model. By employing positive-definite estimators, the sampling error at each site can be rigorously bounded relative to the mean via the Markov inequality, overcoming the lack of self-averaging and enabling accurate estimates even under strong spatial fluctuations. The approach extends to non-diagonal observables, such as local currents, through local unitary transformations and its effectiveness is showcased by benchmark calculations in the disordered two-dimensional (2D) $\pi$-flux model, where the LDoS and steady-state current maps are computed. This method will enable simulations of disorder-driven mesoscopic phenomena in realistically large lattices and accelerate real-space self-consistent mean-field calculations.