6 problems

Eternal augmented multiplicative-coalescent conjecture. Fix a $(\kappa,\tau,{\bf c})\in\mathcal{I}$. Then

Let $cP(cell^2_\downarrow)$ be the space of probability measures on $cell^2_\downarrow$, and let $\mathscr R=\mathscr R_{d,L,\alpha}$ be the renormalization map obtained by indepen…

Let $(\boldsymbol{x}^{\scriptscriptstyle(n)})_{n\geqslant 1}$ satisfy Condition II$\tau$ for some $\tau\in(3,4)$, and let $M_n$ be as in the leader theorem. Write $\mathcal{W}_n$ f…

Let $\kappa\geq 0$, $\tau\in\mathbb{R}$, and $\mathbf{c}\in\ell^3$ have non-increasing components. For each $t\in\mathbb{R}$, let $\mathbf{X}(t)$ be the ordered excursion lengths a…

Uniqueness conjecture. For any such function $g_0$, there is a unique measure $\nu$ such that $g_t(x)$, controlled by $\Phi(t)=\nu[0,t]$, satisfies

The mean field forest fire model has big components whose dynamics, after critical scaling, are described by a coagulation-fragmentation process. Let the lightning rate be…