Cluster scaling and critical points: A cautionary tale

Many systems in nature are conjectured to exist at a critical point, including the brain and earthquake faults. The primary reason for this conjecture is that the distribution of clusters (avalanches of firing neurons in the brain or regions of slip in earthquake faults) can be described by a power law. Because there are other mechanisms such as 1/f noise that can produce power laws, other criteria that the cluster critical exponents must satisfy can be used to conclude whether or not the observed power-law behavior indicates an underlying critical point rather than an alternate mechanism. We show how a possible misinterpretation of the cluster scaling data can lead one to incorrectly conclude that the measured critical exponents do not satisfy these criteria. Examples of the possible misinterpretation of the data for one-dimensional random site percolation and the one-dimensional Ising model are presented. We stress that the interpretation of a power-law cluster distribution indicating the presence of a critical point is subtle and its misinterpretation might lead to the abandonment of a promising area of research.

Automated summary

Many natural systems are believed to exist at a critical point, characterized by a power law distribution of clusters. However, since power laws can also be produced by other mechanisms, it is important to use additional criteria to determine whether the observed power-law behavior indicates a critical point or an alternate mechanism. This study demonstrates how misinterpretation of cluster scaling data can lead to the incorrect conclusion that the measured critical exponents do not satisfy these criteria, emphasizing the significance of properly interpreting power-law cluster distributions in order to avoid abandoning promising research areas.