Tails of the crossing probability

Area of Science:

• Statistical Physics
• Complex Systems
• Computational Physics

Background:

• Percolation theory describes the formation of connected clusters in random systems.
• Correlated percolation models, like the q-state Potts model, introduce dependencies between sites and bonds.
• Understanding anisotropic percolation (horizontal-only) is crucial for various physical phenomena.

Purpose of the Study:

• To numerically investigate the scaling of horizontal-only percolation probability (pi(hs)) in the correlated site-bond percolation model.
• To analyze the behavior of pi(hs) far from and at the critical point.
• To identify and characterize the scaling indices governing these behaviors.

Main Methods:

• Numerical simulations of the correlated site-bond percolation model (q-state Potts model) for q=1, 2, 3, 4.
• Analysis of the crossing probability pi(hs) as a function of system parameters near the critical point.
• Characterization of scaling functions and determination of critical exponents.

Main Results:

• Demonstrated a specific exponential scaling form for pi(hs) far from the critical point: pi(hs)(p) ~ D exp[cL(p-p(c))^nu].
• Observed a crossover to a different scaling form at criticality: pi(hs)(p) ~ A exp{-b[L(p-p(c))^nu]^x}.
• Identified nu as the correlation length index and x as a scaling index for the central crossing probability.

Conclusions:

• The study elucidates distinct scaling regimes for horizontal-only percolation in correlated systems.
• The findings provide insights into critical phenomena and universality in percolation models.
• The identified scaling indices contribute to a deeper understanding of anisotropic cluster formation.