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Area of Science:

  • Statistical Mechanics
  • Probability Theory
  • Complex Systems

Background:

  • The beta random walk is a discrete-time model on a 1D lattice.
  • It features space- and time-dependent jump probabilities.
  • Understanding its large-deviation properties is crucial for complex systems.

Purpose of the Study:

  • To analyze the large-deviation rate function of the beta random walk.
  • To compare its rate function with the continuum version of the model.
  • To validate analytical predictions through numerical simulations.

Main Methods:

  • Analytical derivation of the large-deviation rate function.
  • Numerical simulations using an importance sampling algorithm.
  • Comparison between discrete and continuum models.

Main Results:

  • The large-deviation rate function of the beta random walk closely matches its continuum counterpart.
  • Numerical simulations confirm analytical predictions with high accuracy.
  • A first-order phase transition in the tilted measure was observed.

Conclusions:

  • The beta random walk exhibits universal large-deviation behavior shared with its continuum analog.
  • The study validates theoretical predictions and demonstrates the applicability of macroscopic fluctuation theory.
  • This research provides insights into heat transfer models like the KMP model.