A simple unified view of branching process statistics: random walks in balanced logarithmic potentials

Abstract

We revisit the problem of deriving the mean-field values of avalanche critical exponents in systems with absorbing states. These are well-known to coincide with those of an un-biased branching process. Here, we show that for at least 4 different universality classes (directed percolation, dynamical percolation, the voter model or compact directed percolation class, and the Manna class of stochastic sandpiles) this common result can be obtained by mapping the corresponding Langevin equations describing each of these classes into a random walker confined close to the origin by a logarithmic potential. Many of the results derived here appear in the literature as independently derived for individual universality classes or for the branching process. However, the emergence of non-universal continuously-varying exponent values --which, as shown here, stems fro the presence of small external driving, that might induce avalanche merging-- has not been noticed (or emphasized) in the past. We believe that a simple an unified perspective as the one presented here can (i) help to clarify the overall picture, (ii) underline the super-universality of the behavior as well as the dependence on external driving, and (iii) help avoiding the common existing confusion between un-biased branching processes (equivalent to a random walker in a balanced logarithmic potential) and standard (un-confined) random walkers.

arXiv comment6 pages, 3 figures, 1 table

References in corpus (21)

• Crackling Noise
• Critical Coarsening without Surface Tension: the Voter Universality Class
• Universal scaling behavior of non-equilibrium phase transitions
• Self-Organized Branching Processes: A Mean-Field Theory for Avalanches
• Directed percolation criticality in turbulent liquid crystals
• Beyond power laws: Universality in the average avalanche shape
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Avalanche and spreading exponents in systems with absorbing states
• Driving, conservation and absorbing states in sandpiles
• Exactly solvable model of avalanches dynamics for Barkhausen crackling noise
• Self-organization without conservation: true or just apparent scale-invariance?
• Langevin description of critical phenomena with two symmetric absorbing states
• Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
• Phase Structure of Systems with Multiplicative Noise
• The average shape of a fluctuation: universality in excursions of stochastic processes
• Avalanche shape and exponents beyond mean-field theory
• Hyperscaling in the Domany-Kinzel Cellular Automaton
• Mean-field limit of systems with multiplicative noise
• Nonequilibrium transitions induced by multiplicative noise
• Universality and a numerical ε-expansion of the Abelian Manna model below upper critical dimension
• Mapping multiplicative to additive noise

Cited by in corpus (28)

• Inferring change points in the COVID-19 spreading reveals the effectiveness of interventions
• Colloquium: Criticality and dynamical scaling in living systems
• Landau-Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization
• Hysteresis, neural avalanches and critical behaviour near a first-order transition of a spiking neural network
• Neutral theory and scale-free neural dynamics
• Hybrid-type synchronization transitions: where marginal coherence, scale-free avalanches, and bistability live together
• Growing Critical: Self-Organized Criticality in a Developing Neural System
• Time-series thresholding and the definition of avalanche size
• Theoretical foundations of studying criticality in the brain
• Hopf Bifurcation in Mean Field Explains Critical Avalanches in Excitation-Inhibition Balanced Neuronal Networks: A Mechanism for Multiscale Variability
• Self-organized bistability and its possible relevance for brain dynamics
• First-passage distributions for the one-dimensional Fokker-Planck equation
• Disentangling the critical signatures of neural activity
• Critical branching processes in digital memcomputing machines
• Non-normality, reactivity, and intrinsic stochasticity in neural dynamics: a non-equilibrium potential approach
• Classes of critical avalanche dynamics in complex networks
• Anomalous Diffusion: Fractional Brownian Motion vs. Fractional Ito Motion
• Evolution in the Debian GNU/Linux software network: analogies and differences with gene regulatory networks
• Jensen's force and the statistical mechanics of cortical asynchronous states
• Emergent spatial patterns of coexistence in species-rich plant communities
• Power-law and log-normal avalanche size statistics in random growth processes
• Percolation theory of self-exciting temporal processes
• Critical growth of cerebral tissue in organoids: theory and experiments
• Characterizing spatial point processes by percolation transitions
• Critical avalanches of Susceptible-Infected-Susceptible dynamics in finite networks
• Dynamics of a Bottom-Heavy Janus Particle Near a Wall Under Shear Flow
• The Limits of Inference in Complex Systems: When Stochastic Models Become Indistinguishable
• Higher-order contagion processes in 3.99 dimensions