5 results found
Garcia Millan R, Pausch J, Walter B, et al., 2018, Field-theoretic approach to the universality of branching processes, Physical Review E, Vol: 98, ISSN: 1539-3755
Branching processes are widely used to model phenomena from networks to neuronal avalanching. In a large class of continuous-time branching processes, we study the temporal scaling of the moments of the instant population size, the survival probability, expected avalanche duration, the so-called avalanche shape, the n-point correlation function, and the probability density function of the total avalanche size. Previous studies have shown universality in certain observables of branching processes using probabilistic arguments; however, a comprehensive description is lacking. We derive the field theory that describes the process and demonstrate how to use it to calculate the relevant observables and their scaling to leading order in time, revealing the universality of the moments of the population size. Our results explain why the first and second moment of the offspring distribution are sufficient to fully characterize the process in the vicinity of criticality, regardless of the underlying offspring distribution. This finding implies that branching processes are universal. We illustrate our analytical results with computer simulations.
Garcia Millan R, Pruessner G, Pickering L, et al., 2018, Correlations and hyperuniformity in the avalanche size of the Oslo Model, Europhysics Letters: a letters journal exploring the frontiers of physics, Vol: 122, ISSN: 1286-4854
Certain random processes display anticorrelations resulting in local Poisson-like disorder and global order, where correlations suppress fluctuations. Such processes are called hyperuniform. Using a map to an interface picture we show via analytic calculations that a sequence of avalanche sizes of the Oslo model is hyperuniform in the temporal domain with the minimal exponent $\lambda=0$ . We identify the conserved quantity in the interface picture that gives rise to the hyperuniformity in the avalanche size. We further discuss the fluctuations of the avalanche size in two variants of the Oslo model. We support our findings with numerical results.
Corral A, Garcia-Millan R, Moloney NR, et al., 2018, Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects, PHYSICAL REVIEW E, Vol: 97, ISSN: 2470-0045
Corral A, Garcia-Millan R, Font-Clos F, 2016, Exact derivation of a finite-size scaling law and corrections to scaling in the geometric galton-watson process, PLoS ONE, Vol: 11, Pages: 1-17, ISSN: 1932-6203
The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.
Garcia-Millan R, Font-Clos F, Corral A, 2015, Finite-size scaling of survival probability in branching processes, PHYSICAL REVIEW E, Vol: 91, ISSN: 2470-0045
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