Markov Inequality as a Tool for Linear-Scaling Estimation of Local Observables
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918205219405824 |
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| author | Veiga, H. P. Pinheiro, D. R. Pires, J. P. Santos Lopes, J. M. Viana Parente |
| author_facet | Veiga, H. P. Pinheiro, D. R. Pires, J. P. Santos Lopes, J. M. Viana Parente |
| contents | 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) $π$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21688 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Markov Inequality as a Tool for Linear-Scaling Estimation of Local Observables Veiga, H. P. Pinheiro, D. R. Pires, J. P. Santos Lopes, J. M. Viana Parente Disordered Systems and Neural Networks Mesoscale and Nanoscale Physics 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) $π$-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. |
| title | Markov Inequality as a Tool for Linear-Scaling Estimation of Local Observables |
| topic | Disordered Systems and Neural Networks Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2510.21688 |