Garcia-Millan Rosalba, Font-Clos Francesc, Corral Álvaro
Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, E-08193 Barcelona, Spain.
Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, E-08193 Barcelona, Spain.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):042122. doi: 10.1103/PhysRevE.91.042122. Epub 2015 Apr 20.
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.
分支过程在统计物理学的许多模型中普遍存在。我们研究了高尔顿 - 沃森分支过程在有限代数后的生存概率。我们通过分析得出了生存概率作为控制参数和最大代数函数的有限尺寸标度的存在性，得到了临界指数以及精确的标度函数，即(G(y)=2ye(y)/(e(y)-1))，其中(y)是到临界点的重标距离。我们的发现对于任何高尔顿 - 沃森类型的分支过程都是有效的，与后代数量的分布无关，只要其方差是有限的。这证明了分支过程中有限尺寸效应的普遍行为，包括度量因子的普遍性。还讨论了与平均场渗流的直接关系。
