Random growth via gradient flow aggregation

Abstract We introduce gradient flow aggregation, a random growth model. Given existing particles $\{x_1,\ldots,x_n\} \subset \mathbb{R}^2$ , a new particle arrives from a random direction at $\infty$ and flows in direction of the vector field $\nabla E$ where $ E(x) = \sum_{i=1}^{n}{1}/{\|x-x_i\|^{\alpha}}$ , $0 < \alpha < \infty$ . The case $\alpha = 0$ refers to the logarithmic energy ${-}\sum\log\|x-x_i\|$ . Particles stop once they are at distance 1 from one of the existing particles, at which point they are added to the set and remain fixed for all time. We prove, under a non-degeneracy assumption, a Beurling-type estimate which, via Kesten’s method, can be used to deduce sub-ballistic growth for $0 \leq \alpha < 1$ , $\text{diam}(\{x_1,\ldots,x_n\}) \leq c_{\alpha} \cdot n^{({3 \alpha +1})/({2\alpha + 2})}$ . This is optimal when $\alpha = 0$ . The case $\alpha = 0$ leads to a ‘round’ full-dimensional tree. The larger the value of $\alpha$ , the sparser the tree. Some instances of the higher-dimensional setting are also discussed.