6 problems

Stationary-measure and phase-diagram conjecture. The stated family gives all extremal stationary measures, and the three parameter regimes give the corresponding weak limits of…

Let $(P_n)_{n\geq 0}$ be the suitably scaled maximal entropy random walk pyramidal process. Define … and let $\mathcal{H}$ be the graph of $\Omega$. Set … and…

Let $\mathbb{C}_0(t)$ denote the relevant boundary cluster at time $t$, and let $N$ be the scaling parameter used for the Poisson percolation model. Fluctuation conjecture. Fluctua…

Let $(A_n)_{n\geq 0}$ be the DLA cluster process in the wedge $W_{\theta_1,\theta_2}$, and define its growth rate by … where $\operatorname{diam}$ denotes Euclidean diameter, and l…

Let $A_\infty$ be the limiting diffusion-limited aggregation cluster in a wedge, and interpret its arms as the ends of the graph $A_\infty$. One-arm conjecture. There exists…

Let $\widehat{R}$ denote the front of the modified system in which, when multiple absorptions are possible, exactly one particle is absorbed, the front advances one step, and all e…