25 problems

Asymptotic shape conjecture. There exists a nonempty compact subset $B_0$ of $\mathbb{R}^2$ such that, for every positive number $\varepsilon$,

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…

Let $G$) be a graph with distinguished vertex $x$, and let $H=G\times\{0,1\}$ have two copies of $G$, with corresponding vertices joined by an edge. Run Richardson's model on $H$ f…

Derivative probability-ratio conjecture. There exists a constant $\theta\in(0,d/2)$ such that

Positive-drift conjecture. There exists $\theta_k\in(0,1)$ such that, for most reasonable $k$-edge sets $S$ and every positive $m$,

Nonnegative-expectation conjecture. For most sets $S$, the expected value of the higher-order derivative is non-negative:

Second-order mass conjecture. There exists a constant $C$ independent of $N$ such that

Comparability conjecture. There exists a constant $C$ independent of $n$ such that

Let $T_n(1,2)$ be the passage time between vertices $1$ and $2$ in the critical random graph model, and suppose the edge passage time has distribution … Thus each edge has passage…

Let $a\from (3/2,5/2)$, let $\bigl(f_i\bigr)_{i\geq 1}$ be the sampled marked vertices or faces, let $d^\dagger_\mathrm{fpp}$ denote the first-passage percolation distance, and let…

Consider the long-range percolation graph on $\mathbb T_n^d$ with all nearest-neighbour edges and each non-nearest-neighbour edge included independently with probability…

Consider the competing long-range infection model on $\mathbb T_n^d$ with long-range parameters $\alpha_\ominus(n)$ and $\alpha_\oplus(n)$, infection-rate ratio $\lambda(n)$, and c…

Consider the competing long-range infection model and the notation of the theorem concerning absence of coexistence, including the final-size quantity $M_n$ and the parameter $Z_n$…

Let $T_h$ be the limiting passage time defined by the preceding convergence result, let $x\in\mathbb R$ be fixed, and let $\mathcal A(x)$ denote the stationary Airy process. Airy-p…

Survival-regime conjecture.

Barrier-cone characterization conjecture. For every $p\in[0,1]$,

Let $\gamma_{0}^\star=(0,v_1^\star,v_2^\star,\dots):=\arg\min\{\lvert\gamma_L\rvert\}$ be the shortest exploding path starting at $0$, where $\lvert\gamma_L\rvert$ denotes its $L$-…

Let $d_L(x,y)$ denote the $L$-distance between vertices $x,y\in\mathbb{Z}^d$, and let $V_x$ be the explosion time from $x$. Under the condition of the distance theorem, these quant…

Let $(t_e)$ be i.i.d. edge weights on the nearest-neighbor edges of $\mathbb{Z}^2$, with a continuous distribution. A bigeodesic is a doubly infinite path whose finite segments are…

Let $F_a$ be the distribution defined by … where $p_c$ is the critical probability for Bernoulli bond percolation on $\mathbb{Z}^2$, and let…

Sublinear variance conjecture. The estimate $$ is satisfied for all Cayley graphs of $G$.

Let $B_g(t)$ be the random metric ball at time $t$, and let $B(t/\mu)$ be the corresponding rescaled limiting shape, where $\mu$ is the time constant. The fluctuations of these two…

Let $B^{\mathrm{E}}_n=B^{\mathrm{E}}(0,n)$ be the Euclidean ball of radius $n$, let $\mathcal V_n\subseteq S^{d-1}$ be the set of initial velocities of minimizing unit-speed geodes…

Let $a_n(h)$ denote the first-passage percolation time across a cylinder of length $n$ and half-height $h$, and let $d\ge 2$. Thin-cylinder variance-order conjecture. The variance…

Let $a_n(h)$ be the first-passage percolation time across a cylinder of length $n$ and half-height $h$, and define … For distributions satisfying the stated moment and admissibilit…