6 problems

Let $\beta_i(t)$ be the $i$th Betti number of the $d$-dimensional Eden growth model at time $t$, with $1\leq i\leq d-1$. Almost-sure scaling conjecture. There exists a constant…

Let $n$ be the number of pentagons in the model, and let $V(n)$, $E(n)$, and $H(n)$ denote the numbers of vertices, edges, and holes, respectively. The expectation is taken over th…

Let $P_0$ be the original pentagon in the Eden model, and let $u_1,u_2,u_3,u_4,u_5$ be the five possible translation vectors between centers of neighboring pentagons. The set of po…

Ballistic deposition height-growth conjecture. With probability one, for all $j\in\mathbb N$,

Let $X_N^R$ denote the top line of an Aztec diamond of size $N$ with a triangle removed from the top corner at height $N/\sqrt{2}+2^{-5/6}N^{1/3}R$. Define … For given observation…

Let $X_N^R$ denote the top line of an Aztec diamond of size $N$ with a square removed from the top so that its lower tip is at height $N/\sqrt{2}+2^{-5/6}N^{1/3}R$. Define … For an…