Universality of percolation at dynamic pseudocritical point

TL;DR

This paper investigates the universal behaviors of percolation models at pseudocritical points, revealing that certain distribution forms and dimensionless quantities exhibit quasi-universality across different lattices and models.

Contribution

It demonstrates that at pseudocritical points, the size distribution of the largest cluster follows universal forms and certain dimensionless quantities are nearly universal, extending the understanding of universality beyond the critical point.

Findings

Largest cluster size distribution approaches Gumbel or Gaussian forms in different phases.

Dimensionless quantities like wrapping probabilities are nearly universal at pseudocritical points.

Critical polynomial is nonzero at pseudocritical points, unlike at the critical point.

Abstract

Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.