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Markov inequality as a tool for linear-scaling estimation of local observables

H. P. Veiga1,2,*,†, D. R. Pinheiro1,*, J. P. Santos Pires1, and J. M. Viana Parente Lopes1,‡

  • *These authors contributed equally to this work.
  • †Contact author: henriqueveigacj@gmail.com
  • ‡Contact author: jlopes@fc.up.pt
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Phys. Rev. Research 8, 033015 – Published 6 July, 2026

DOI: https://doi.org/10.1103/qb1w-44r1

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 semidefinite 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 nondiagonal observables, such as local currents, through local unitary transformations and its effectiveness is showcased by benchmark calculations in the disordered two-dimensional π-flux model, where the local density of states and steady-state current maps are computed. This method will enable simulations of disorder-driven mesoscopic physics in realistically large lattices and accelerate real-space self-consistent mean-field calculations.

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Phys. Rev. Research 8, 033015– Published 6 July, 2026
  • Received 20 November 2025
  • Accepted 1 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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