Localized growth and branching random walks with time correlations

We generalize a model of growth over a disordered environment, to a large class of Itō processes. In particular, we study how the microscopic properties of the noise influence the macroscopic growth rate. The present model can account for growth processes in large dimensions and provides a bed to understand better the tradeoff between exploration and exploitation. An additional mapping to the Schrödinger equation readily provides a set of disorders for which this model can be solved exactly. This mean-field approach exhibits interesting features, such as a freezing transition and an optimal point of growth, which can be studied in detail, and gives yet another explanation for the occurrence of the Zipf law in complex, well-connected systems.

Figure 1

(Sketch) Illustration of the mean-field model of exploration and exploitation. (a) Polymers on tree (or branching Brownian motion), a model developed in Ref. [13] for tackling the spin-glass problem. (b) Mean-field model of population growth: sites undergo random phase of multiplicative growth or migration. The migration is diffusion-like: sites with large population spread the excess of them (the green block) over a time scale $\tau =1/\lambda$, while conserving the total number of agents. This amounts to a discrete, infinite-range Laplacian of strength $\lambda$.

Figure 2

(Sketch) The growth rate $c_{\lambda }$ as a function of $\gamma$. Typically, $c_{\lambda }\left(\gamma \right)$ decreases at small $\gamma$ and increases at large $\gamma$. For a front prepared with an initial decay ${\gamma }_{t=0}<{\gamma }_{\text{min}}\left(\lambda \right)$, the front propagates at $c_{\lambda }\left({\gamma }_{t=0}\right)$. On the other hand, for a front with ${\gamma }_{t=0}>{\gamma }_{\text{min}}\left(\lambda \right)$ (the increasing branch of $c_{\lambda }\left(\gamma \right)$), the front will relax toward the minimum, asymptotically propagating at a speed $c_{\lambda }\left({\gamma }_{\text{min}}\right)$ (the frozen regime).

Figure 3

(Sketch) The growth rate $c\left(\lambda \right)$ as a function of $\lambda$ in log-log plot. Once the critical diffusion rate ${\lambda }_c$ is fixed by ${\gamma }_{\text{min}}\left({\lambda }_c\right)=1$, it separates two regimes corresponding to quenched (dashed-dot, $\lambda <{\lambda }_c$) and annealed (dashed, $\lambda >{\lambda }_c$) solutions of Eq. (28). Both branches touch at ${\lambda }_c$. In large (red) dashes represent the full curve as dictated by the Kolmogorov velocity selection principle, for a noise with a nonzero correlation time. The white-noise case is also depicted, with black squares, and reaches a plateau equal to $|{\alpha }_0(\gamma =1)|$ at ${\lambda }_c$. Typical growth rates exhibit a maximum at a value ${\lambda }_m$, in the quenched phase ${\lambda }_m<{\lambda }_c$.

Figure 4

$c\left(\lambda \right)$ as a function of $\lambda$ for an Ornstein-Uhlenbeck process, with the set of parameters (from top to bottom): $D_2=5000$, $k=100$; $D_2=50$, $k=10$; $D_2=1$, $k=\sqrt{2}$; $D_2=0.5$, $k=1$. The dashed line is the expansion Eq. (36). The scaling have been chosen so that the upper bound $|{\alpha }_0(\gamma =0)|$ is fixed to the value 0.5. The numerics are performed on a system of size $N={10}^6$ sites, up to a time $t_{\text{tot}}=500$, with the discretization time step $dt=0.001$.

Figure 5

$c\left(\lambda \right)$ as a function of $\lambda$ for the bounded process, with the set of parameters (from top to bottom): $D_2$ and $a$ fitted from Eq. (39); $D_2=0.5$, $a=1$; $D_2=1$, $a=1$. The dotted line is a fit obtained from the OU process, matching the asymptotic behavior. The dashed line is the expansion Eq. (36). The numerics are performed on a system of size $N={10}^6$ sites, up to a time $t_{\text{tot}}=500$, with the discretization time step $dt=0.001$.

Figure 6

Scaling of ${\lambda }_c$ for the OU process (left) Scaling of ${\lambda }_c$ with $T$, and $\langle {\eta }^2\rangle =1/\sqrt{2}$. (Right) Scaling of ${\lambda }_c$ with $\langle {\eta }^2\rangle$, and $T=1.0$. Both dashed lines are guidelines of unit slopes.

Figure 7

Scaling of the difference ${\lambda }_c-{\lambda }_m$ for the OU process. We have fixed $\epsilon =\frac{D_2}{k^2}=\frac{1}{2}$. The blue dots are the result of the numerical solution of Eq. (38) for specific values of $D_2$ and $k=1/T$, while the black dashed line is a small $T$ expansion: ${\lambda }_c-{\lambda }_m$ widens linearly with $T$, close to $T=0$.