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How fast does a random walk cover a torus?

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High statistics simulations confirm random walk covering time scaling on a 2D torus. However, the prefactor deviates from theoretical predictions, suggesting slow convergence for lattice walks.

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Area of Science:

  • Probability theory
  • Statistical physics
  • Computational mathematics

Background:

  • The time for a random walk to visit all sites on a lattice (cover time) is a fundamental problem in probability.
  • Previous theoretical work predicted a specific scaling law for the cover time on a 2D torus.

Purpose of the Study:

  • To present high statistics simulation data for the cover time of a random walk on a 2D torus.
  • To compare simulation results with existing theoretical predictions, particularly the prefactor of the scaling law.
  • To investigate the convergence of cover times and reconcile potential discrepancies between theory and simulation.

Main Methods:

  • High statistics numerical simulations of random walks on L×L two-dimensional tori.
  • Analysis of the average cover time 〈T_{cover}(L)〉 and the time T_{N(t)=1}(L) when only one site remains unvisited.
  • Comparison of simulation data with theoretical predictions, including scaling laws and prefactors.

Main Results:

  • Simulations confirm the predicted scaling 〈T_{cover}(L)〉∼(LlnL)^{2} for large L.
  • A significant deviation in the prefactor from the theoretically predicted 4/π was observed using straightforward extrapolation.
  • The scaling law holds for T_{N(t)=1}(L), and the distribution of rescaled cover times sharpens as L→∞.
  • Results are reconciled with Dembo et al.'s work by considering a slow, nonmonotonic convergence of the prefactor for lattice walks.

Conclusions:

  • The study validates the asymptotic scaling of random walk cover times on a 2D torus.
  • Discrepancies in the prefactor highlight the importance of slow convergence effects in lattice models.
  • The findings support conjectures about the behavior of cover times for lattice walks, aligning with Brownian motion results.