6 problems

Monotonicity conjecture. For $\lambda<\lambda'\in(0,1]$,

Häggström–Pemantle's conjecture. In every dimension $d\geq 2$, infinite coexistence has positive probability if and only if the infection intensities are equal:

Let $d\geq 2$. For $i=1,2$, let $A_i$ be the event that infection type $i$ reaches sites arbitrarily far from the origin, and let $A=A_1\cap A_2$ be the coexistence event. Let…

Let $R(t)$ and $B(t)$ be the red and blue regions of the two-species competition model, and let $\mathcal{S}$ be the Richardson-model limit shape. For a subset…

Consider the competition process on $\mathbb{Z}^d$ with blue and yellow infections, whose transmission parameters are $p_{\text{b}}$ and $p_{\text{y}}$, respectively. Let $p_c$ be…

Let $v(\cdot)$ be the speed function of a reasonable $(2+1)$-dimensional growth model, and let $H_\rho$ be the Hessian of $v(\cdot)$ computed at the slope $\rho$. Write $\alpha$ fo…