150 problems

In the two-type Richardson model on ${\mathbb Z}^d$, let $G_i$ be the event that type $i$ infects sites arbitrarily far from the origin, and let $G=G_1\cap G_2$. After rescaling ti…

For $d=1$, minimal surfaces on the infinite domain $Lambda_\infty=\mathbb{Z}$ are analogous to bigeodesics in first-passage percolation. Bigeodesic nonexistence conjecture. No bige…

In two-type first passage percolation on the lattice, two infection types spread simultaneously using passage times with potentially different intensities. The event that both infe…

Let $Gamma(0)subseteqmathbb{E}$ be the graph of infection from the origin in first-passage percolation, and let $K(Gamma(0))$ denote its number of ends. **Newman's infinite-ends c…

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…

Differentiability and strict convexity conjecture. If the edge-weight distribution is not deterministic, then the limit shape is differentiable; if the distribution is continuous,…

Let $T_k^n$ denote the relevant point-to-point passage time, and let $\operatorname{Var}(T_k^n)$ be its variance. Variance conjecture. The variance satisfies … This prediction is u…

Eden model limit-shape conjecture. For every dimension $d\geq 2$, the limit shape of the Eden model is not a Euclidean ball.

Cox–Durrett conjecture. The limiting time constant is

Let $\mathcal{G}_{n,r}$ be a random $r$-regular graph with independent exponential edge weights, and let $\operatorname{Diam}_w(\mathcal{G}_{n,r})$ denote its weighted diameter. Se…

Consider the two-type Richardson model with infection rates $\lambda_1$ and $\lambda_2$, and let $A$ denote the event that both types grow indefinitely from their finite initial co…

Let $\Gamma$ be the infection tree on $\mathbb{Z}^2$ rooted at the origin, whose edges are the edges on fastest paths from the origin to each vertex, and let $K(\Gamma)$ be the num…

Outside-cone fluctuation conjecture. If

Axial curvature and fluctuation exponent conjecture. For $r>0$,

Transversal fluctuation exponent conjecture. The observation that the oriented-percolation example rules out the assertion $\xi(u)=2/3$ for every $u$ shows that this proposed unive…

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…

KPZ fluctuation exponent conjecture. The right order of the fluctuations of $T(\0,{\mathbf n})-\mu n$ is $n^{\chi}$, where

Let $T$ denote the first-passage percolation passage time, with expectation $\mathbb{E}[T]$ and variance $\operatorname{var}(T)$. The two-thirds variance exponent conjecture. The v…

Let $p<p_c(2)$, where $p_c(2)$ is the critical probability for Bernoulli bond percolation on ${\bf Z}^2$. Let $B_2$ be the unit ball of the time-constant norm, let…

Let $H^{\alpha}({\mathbf x},{\mathbf y})$ be the Busemann function in direction $e^{i\alpha}$, defined from the passage-time difference between the vertices associated with…

Newman's conjecture. For a large class of stationary measures ${\mu}$,

Consider first-passage percolation on the two-dimensional integer lattice $\mathbf{Z}^2$, with independent identically distributed passage times that are exponential random variabl…

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…

Let $\xi=\xi(d)$ denote the wandering exponent, defined so that $|y-x|^\xi$ gives the typical or largest order of deviation of a geodesic from the straight line segment…