title: Exact Derivation of a Finite-Size Scaling Law and Corrections to Scaling in the Geometric Galton-Watson Process.
abstract: 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.
authors: Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain.|Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, E-08193 Barcelona, Spain.
authors: Álvaro
authors: Á
authors: Corral
authors: London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom.|Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, United Kingdom.
authors: Rosalba
authors: R
authors: Garcia-Millan
authors: ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy.
authors: Francesc
authors: F
authors: Font-Clos
mesh_terms: D008962:Models, Theoretical; D013223:Statistics as Topic / Q000379:methods*; D013269:Stochastic Processes; D013816:Thermodynamics