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  • Open Access

Numerical estimation of limiting large-deviation rate functions

Peter Werner* and Alexander K. Hartmann†

  • *Contact author: peter.werner@uni-oldenburg.de
  • †Contact author: a.hartmann@uni-oldenburg.de
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Phys. Rev. E 113, 034117 – Published 13 March, 2026

DOI: https://doi.org/10.1103/sj6t-pctp

Abstract

For statistics of rare events in systems obeying a large-deviation principle, the rate function is a key quantity. When numerically estimating the rate function, one is always restricted to finite system sizes. Thus, if the interest is in the limiting rate function for infinite system sizes, first, several system sizes have to be studied numerically. Here, rare-event algorithms using biased ensembles give access to the low-probability region. Second, some kind of system-size extrapolation has to be performed. Here, we demonstrate how rare-event importance sampling schemes can be combined with multihistogram reweighting. We study two ways of performing the system-size extrapolation, either directly acting on the empirical rate functions or on the scaled cumulant generating functions, to obtain the infinite-size limit. The presented method is demonstrated for a binomial distributed variable, a Markov process of random bits, and the largest connected component of Erdős-Rényi random graphs. Analytical solutions are available in all cases for direct comparison. It is observed in particular that phase transitions appearing in the biased ensembles can lead to systematic deviations from the true result.

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Phys. Rev. E 113, 034117– Published 13 March, 2026
  • Received 13 November 2025
  • Accepted 23 February 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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