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Numerical estimation of limiting large-deviation rate functions.

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This study combines rare-event algorithms with multihistogram reweighting to estimate the limiting rate function for large systems. The method accurately extrapolates finite-size numerical results to the infinite-size limit.

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Area of Science:

  • Statistical physics
  • Computational methods
  • Probability theory

Background:

  • Estimating the rate function for rare events in large systems is crucial for statistical analysis.
  • Numerical estimations are limited to finite system sizes, necessitating extrapolation to infinite sizes.
  • Rare-event algorithms and biased ensembles provide access to low-probability regions.

Purpose of the Study:

  • To develop and demonstrate a method for extrapolating finite-size numerical estimates of rate functions to the infinite-size limit.
  • To combine rare-event importance sampling with multihistogram reweighting for accurate system-size extrapolation.
  • To compare different extrapolation strategies acting on rate functions versus scaled cumulant generating functions.

Main Methods:

  • Utilizing rare-event importance sampling to explore low-probability regions.
  • Applying multihistogram reweighting to combine data from different system sizes.
  • Performing system-size extrapolation directly on empirical rate functions or scaled cumulant generating functions.
  • Validating the method against analytical solutions for benchmark systems.

Main Results:

  • The combined approach successfully estimates the infinite-size limit of the rate function.
  • Two distinct system-size extrapolation techniques were investigated and compared.
  • The method was demonstrated on a binomial variable, a Markov process, and Erdős-Rényi random graphs.
  • Phase transitions in biased ensembles can introduce systematic deviations.

Conclusions:

  • The proposed method provides a robust way to obtain the infinite-size rate function from finite-size simulations.
  • Combining rare-event sampling with reweighting techniques is effective for large-deviation statistics.
  • Care must be taken to account for potential artifacts arising from biased ensembles.