Scaling theory and spreading dynamics in systems with one absorbing state derived from an equilibrium statistical model.
We show that for systems with one absorbing state, the widely assumed scaling properties of the survival probability and of the probability density of the size of activity avalanches cannot be true in the asymptotic limit. Trying to answer the question, what is the true asymptotic limit of these quantities, we study Domany-Kinzel probabilistic cellular automata using an equilibrium statistical mechanic model (ESM). We are able to express important quantities of the avalanche dynamics by correlation functions of the ESM. The application of scaling theory to the ESM allows for the derivation of the scaling properties of quantities of the avalanche dynamics in the form of infinite series. From these results we can obtain possible solutions for the apparent scaling problem, but cannot decide definitely which one is true. The most appealing solution, for which some evidence is given, states that there is a narrow range around the critical point in which, for example, the survival probability has the same power-law behavior as on the critical point. Outside this narrow range, the usually assumed scaling should be approximately valid.
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