Extreme events scaling in self-organized critical models

TL;DR

This work investigates extreme avalanche events in finite-size two-dimensional self-organized critical systems, focusing on the stochastic Manna model and the Bak–Tang–Wiesenfeld sandpile. Using block-maxima and generalized extreme value theory, the authors find that extreme avalanche size follows the Gumbel class with $ξ=0$, while extreme avalanche area exhibits a positive shape parameter $ξ>0$, and they establish finite-size scaling functions to collapse data across system sizes. The results show high-quality GEVD fits ($R^2>0.99$) and yield consistent scaling exponents that relate extremes across length scales, providing a framework to understand extreme SOC dynamics. This methodology offers a principled way to characterize rare events in SOC and could inform analyses of extreme phenomena in related complex systems such as weather and geophysical processes.

Abstract

We study extreme events of avalanche activities in finite-size two-dimensional self-organized critical (SOC) models, specifically the stochastic Manna model (SMM) and the Bak-Tang-Weisenfeld (BTW) sandpile model. Employing the approach of block maxima, the study numerically reveals that the distributions for extreme avalanche size and area follow the generalized extreme value (GEV) distribution. The extreme avalanche size follows the Gumbel distribution with shape parameter $ξ=0$ while in the case of the extreme avalanche area, we report $ξ>0$. We propose scaling functions for extreme avalanche activities that connect the activities on different length scales. With the help of data collapse, we estimate the precise values of these critical exponents. The scaling functions provide an understanding of the intricate dynamics for different variants of the sandpile model, shedding light on the relationship between system size and extreme event characteristics. Our findings give insight into the extreme behavior of SOC models and offer a framework to understand the statistical properties of extreme events.

Figures (4)

• Figure 1: In the SMM, the system scaling for mean, variance, mode and the probability for the mode (inset) of (a) extreme avalanche size (b) extreme avalanche area. The straight line represents the best-fit along with the fitted critical exponents.
• Figure 2: The probability distribution $P(x)$ for (a) extreme avalanche size $x_1$ and (b) extreme avalanche area $x_2$ in case of the SMM for different system sizes $N=L^2$, where $L=2^3, 2^4,....,2^7$. The black dashed line has slope $-1$ which represents $P(M) \sim x^{-1}$.
• Figure 3: In SMM, the data collapse corresponding to Fig. \ref{['Fig-Manna-Prob']} for the probability distribution of rescaled (a) extreme avalanche size $(u_1)$ and (b) extreme avalanche area $(u_2)$.
• Figure 4: In SMM, the data collapse of the CDF corresponds to Fig. \ref{['Fig-Manna-Prob']}. The red dashed line represents the fitted data with Gumbel family $(\xi=0)$ while the black dashed line represents the fitted GEV family with parameters $(\mu, \sigma, \xi)$. In both cases, the goodness of fit is $R^2>0.99$.