Exact order-parameter distribution for critical mean-field percolation and critical aggregation

Area of Science:

• Statistical Physics
• Complex Systems
• Critical Phenomena

Background:

• Mean-field percolation and aggregation are fundamental models in complex systems.
• Both processes exhibit critical behavior characterized by shared critical exponents, belonging to the same universality class.
• Percolation is an equilibrium process, while aggregation is a dynamical critical process.

Purpose of the Study:

• To determine the order-parameter distribution for mean-field percolation at its critical point.
• To compare this distribution with that of a mean-field aggregation process at its critical time.
• To investigate the universality of probability density for order-parameter fluctuations in critical phenomena.

Main Methods:

• Analysis of the order-parameter distribution in mean-field percolation theory.
• Comparison with the order-parameter distribution in mean-field aggregation models.
• Utilizing concepts from universality classes and critical exponents.

Main Results:

• The order-parameter distribution for mean-field percolation at the critical point is identified as the Kolmogorov-Smirnov distribution.
• This distribution is found to coincide with the corresponding distribution for a mean-field aggregation process at the critical time.
• The probability density for order-parameter fluctuations is shown to be universal at the infinite lattice critical point.

Conclusions:

• The Kolmogorov-Smirnov distribution characterizes the order-parameter fluctuations for both equilibrium percolation and dynamical aggregation at their respective critical points.
• This demonstrates that the probability density of order-parameter fluctuations at the critical point of an infinite lattice is universal.
• Universality holds independently of whether the critical process is in thermodynamic equilibrium or is a dynamical critical process.