Phenomenological picture of fluctuations in branching random walks

We propose a picture of the fluctuations in branching random walks, which leads to predictions for the distribution of a random variable that characterizes the position of the bulk of the particles. We also interpret the $1/\sqrt{t}$ correction to the average position of the rightmost particle of a branching random walk for large times $t\gg 1$, computed by Ebert and Van Saarloos, as fluctuations on top of the mean-field approximation of this process with a Brunet-Derrida cutoff at the tip that simulates discreteness. Our analytical formulas successfully compare to numerical simulations of a particular model of a branching random walk.

Figure 1

(a) One realization of the continuous BRW up to time $t=8$. To guide the eye, we also plot the theoretical (truncated) mean position of the boundaries of the BRW, namely $\pm {\overline{X}}_t=\pm 2t\mp \frac{3}{2}lnt$ (dashed lines). (b) Distribution of the particles at times $t=2$, 6, and 8 for this particular realization in bins of size 1 $({log}_{10}$ scale on the vertical axis). We see the bulk building up a smoother (more “deterministic”) distribution as time elapses, while the low-density tails remain noisy. Also, for this realization, the distribution is skewed toward negative values of $x$, due to an accidentally large drift in the initial stages, whose memory is kept throughout the evolution.

Figure 2

$\frac{1}{{\gamma }_0}\langle ln{Y}_t\rangle$ from the numerical solution of the FKPP equation with the “critical” initial condition, as a function of $1/\sqrt{t}$. (The constant term is subtracted.) One sees that it converges to the analytical result Eq. (99) (with $t_0\to +\infty$; straight line) for $t\to +\infty$.

Figure 3

$\frac{1}{{\gamma }_0}ln{\overline{Y}}_t$ from the numerical solution of the branching diffusion equation with a cutoff as a function of $1/\sqrt{t}$. (The constant is subtracted.) Again, the numerical solution converges to the analytical result [Eq. (97); straight line] as $t$ gets large.

Figure 4

Distribution of $f$ for $t=1000$ and two different values of $t_0$. The numerical data (points with statistical error bars and bin width) are compared to Eq. (100) (continuous lines) $({log}_{10}$ scale on the vertical axis).

Figure 5

Moments of $f=ln{Y}_t/Y_{t_0}$ of order 2 to 4 for $t=500$ as a function of $t_0$ in logarithmic scales. The numerical data (full lines) are compared to the analytical calculation in Eq. (102) (dashed lines) $({log}_{10}$ scale on both axes). There are about $4\times {10}^5$ realizations in the statistical ensemble used to perform the averages.

Figure 6

Distribution of $f_Z$ for $t=1000$ and two different values of $t_0$. The numerical data (points with statistical error bars and bin width) are compared to Eq. (B5) (continuous lines). $({log}_{10}$ scale on the vertical axis.)