Random Growth via Gradient Flow Aggregation

TL;DR

Gradient Flow Aggregation (GFA) studies random planar growth where each new particle follows the gradient of the energy $E(x)=\sum_{i=1}^{n} \|x-x_i\|^{-\alpha}$ until it touches the existing cluster. The authors develop an elementary Beurling-type estimate for $0\le\alpha\le 1$ and, using Kesten's method, deduce sub-ballistic growth bounds $\mathrm{diam}\{x_1,\dots,x_n\} \le c_\alpha \; n^{(3\alpha+1)/(2\alpha+2)}$, with optimality at $\alpha=0$ where the cluster resembles a round tree. They also analyze the limiting case $\alpha=\infty$, showing straight-line attachments to the convex hull with hull-vertex probabilities tied to opening angles, and extend the discussion to higher dimensions, where Beurling-type and diameter bounds adapt with a phase-transition intuition at $\alpha=d-2$. The paper provides a rigorous, largely self-contained framework connecting gradient-driven aggregation to Beurling/Kesten-style growth analysis, offering insights into the geometry of growth and potential universality with dielectric breakdown and related models. Overall, the results illuminate how modifying the repulsive potential controls branching structure, sparsity, and growth rates in gradient-based random growth processes.

Abstract

We introduce Gradient Flow Aggregation (GFA), a random growth model. Given a set of existing particles $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$, a new particle arrives from a random direction at $\infty$ and flows in direction $\nabla E$ where $$ E(x) = \sum_{i=1}^{n} \frac{1}{\|x-x_i\|^α} \qquad \mbox{where} ~0 < α< \infty.$$ The case $α= 0$ will refer to the logarithmic energy $- \sum\log \|x-x_i\|$. Particles stop once they are at distance 1 of 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 $0 \leq α< 1$ $$\mbox{diam}(\left\{x_1, \dots, x_n\right\}) \leq c_α \cdot n^{\frac{3 α+1}{2α+ 2}}.$$ This is optimal when $α=0$. The case $α= 0$ leads to a `round' full-dimensional tree. The larger the value of $α$ the sparser the tree. Some instances of the higher-dimensional setting are also discussed.

Figures (12)

• Figure 1: Simulation of Gradient Flow Aggregation (GFA) for $n=1000$ particles with various parameters of $\alpha$.
• Figure 2: A fixed cluster of 40 particles. Gradient descent lines are colored by the color of the particle they eventually attach to (particles colored black were not hit by any gradient descent in the simulation). Likelihoods are determined by gradient flow lines at infinity. As $\alpha$ increases, the exposed endpoints gain more mass.
• Figure 3: A tree grown with $\alpha =2$ and the trajectory of 100 incoming gradient flows (equispaced in angle). Much like in other growth models, there is a tendency to avoid existing valleys.
• Figure 4: GFA for $n=5000$ particles with $\alpha=0$ (left) and $n=2500$ particles with $\alpha =2$. Theorem 3 implies that the structure on the left grows as slowly as possible, $\hbox{diam}(x_1, \dots, x_n) \leq c \sqrt{n}$.
• Figure 5: $n=100$ (left), $n=500$ (middle) and $n=2000$ (right) steps of the evolution of GFA with $\alpha = \infty$ tree.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2: Beurling estimate
• Theorem 3: Growth Bound
• Proposition
• Theorem 4
• Lemma 1
• proof
• proof
• Lemma 2
• proof