Convergence to the Massive Modified Arratia Flow

Let $(\gamma_n)$ and $(\ell_n)$ be sequences in $(0,\infty)$, and let $(N_n)$ be a sequence in $\mathbb{N}^+$. For $u\in[0,1]$ and $t\geq 0$, define

$$Z^{N_n}_{\gamma_n,\ell_n}\left(u,\frac{N_n t}{2}\right).$$

Assume

$$N_n\to\infty,\qquad \ell_n\downarrow 0,\qquad \frac{\gamma_n\ell_n}{\log N_n}\gg 1.$$

Convergence conjecture. Under these assumptions,

$$Z^{N_n}_{\gamma_n,\ell_n}\left(u,\frac{N_n t}{2}\right)\to y(u,t)$$

as $n\to\infty$ in a suitable sense, where $\{y(u,t):u\in[0,1],\ t\in[0,T]\}$ is the Massive Modified Arratia Flow characterized by the stated axioms.

This conjecture predicts that, after clusters form in the regime of strong interaction and short interaction length, the appropriately rescaled weakly interacting diffusion system converges to a system of coalescing heavy Brownian motions represented by the Massive Modified Arratia Flow. The precise mode of convergence is left unspecified in the statement.