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The Use of Chemostats in Microbial Systems Biology
Published on: October 14, 2013
Finite-size scaling of survival probability in branching processes
Rosalba Garcia-Millan1,2, Francesc Font-Clos1,3, Álvaro Corral1,3
1Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, E-08193 Barcelona, Spain.
Abstract:
Branching processes pervade many models in statistical physics. We investigate the survival probability of a Galton-Watson branching process after a finite number of generations. We derive analytically the existence of finite-size scaling for the survival probability as a function of the control parameter and the maximum number of generations, obtaining the critical exponents as well as the exact scaling function, which is G(y)=2ye(y)/(e(y)-1), with y the rescaled distance to the critical point. Our findings are valid for any branching process of the Galton-Watson type, independently of the distribution of the number of offspring, provided its variance is finite. This proves the universal behavior of the finite-size effects in branching processes, including the universality of the metric factors. The direct relation to mean-field percolation is also discussed.
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