Non-equilibrium coagulation processes and subcritical percolation on evolving networks

We investigate percolation on growing networks where the evolution of connected components resembles a non-equilibrium version of the multiplicative coalescent. The supercritical $\pi> \pi_c$ regime for a host of such models was conjectured in statistical physics, and then rigorously proven in mathematics, to exhibit behavior similar to the BKT infinite-order phase transition as $\pi\searrow \pi_c$. It has further been conjectured that the entire regime $\pi<\pi_c$ for such growing networks are ''critical'' with power-law cluster size distributions having a non-universal exponent for all values of $\pi \in (0, \pi_c)$. In this paper, we study percolation on the uniform attachment model, as a concrete template in order to develop general tools based on stochastic approximation, local convergence, branching random walks and tree-graph inequalities to prove the above conjectured phenomena. For each $\pi \in (0,\pi_c)$, we show there exists an explicit $\alpha(\pi) \in (0,\tfrac{1}{2})$ such that the maximal component size, as well as the size of the component containing any fixed vertex, all re-scaled by $n^{\alpha(\pi)}$, converge almost surely to strictly positive random variables as the network size $n \to \infty$. These dynamics lead to novel phenomena, compared to classical 'static' models, including long-range dependence and fixation of the identity of the maximal component, within finite time, among a finite number of 'early' components. Moreover, in contrast with most static network models, we show that the susceptibility, that is, the expected size of the component of a uniformly chosen vertex, remains bounded as the network grows and $\pi$ approaches $\pi_c$ from below. The general tools developed in this paper will be used in follow-up work to understand percolation for general growing network evolution models.