Finite-size scaling of survival probability in branching processes

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 $\mathcal{G}\left(y\right)=2y{e}^y/(e^y-1)$, 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.

Figure 1

Validity of the finite-size scaling law for the geometric Galton-Watson process. (a) Simulation results for the probability of surviving as a function of the mean number of offspring $\langle K\rangle$ in a Galton-Watson process defined by a geometric distribution of $K$, given by $\mathrm{Prob}[K=k]=p{(1-p)}^k$, for $k=0,1,\cdots$ Different values of $L$ show the dependence with system size. Probabilities are estimated from 1000 independent realizations. (b) Same probabilities as a function of $\langle K\rangle -1$ under rescaling with $L$ and ${\sigma }_{\mathrm{c}}$. The collapse of the curves onto a unique scaling function, given by our theoretical result, is the signature of the validity of the finite-size scaling for large system sizes $L$ and close to the critical point $\langle K\rangle =1$.

Figure 2

Universality of the finite-size scaling. Extension of Fig. 1 to other Galton-Watson processes with different distributions for the number of offspring $K$. In addition to the geometric case, the binomial distribution, with $\mathrm{Prob}[K=k]=N!p^k{(1-p)}^{N-k}/[k!(N-k)!]$ and $N=2$ or 3, and the Poisson distribution, with $\mathrm{Prob}[K=k]=e^{-\lambda }{\lambda }^k/k!$, are simulated for different $\langle K\rangle$ and different system sizes $L$. The collapse of all rescaled curves validates the universality of the scaling exponents and of our scaling function.