28 problems

Let $h(t,x)$ be a height function for a model in the KPZ universality class, and let $c_1,c_2,c_3$ be model-dependent constants. Under the KPZ $1:2:3$ scaling, define … Let…

Prähofer–Spohn conjecture. If the limit shape is curved, the limiting process is the Airy$_2$ process; if the limit shape is flat, the limiting process depends further on the rough…

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…

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

Let $\tau\in\mathbb{R}$, let $\mathfrak{h}$ be an upper-semicontinuous function bounded above by a linear function, and define its periodized cutoff by … Let…

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…

The Kardar–Parisi–Zhang universality class contains random growth models in $(1+1)$ dimensions with height functions $H(x,t)$. Let $\mathfrak{h}^{\varepsilon}$ and $\mathfrak{h}$ b…

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…

Start activated random walks on $\mathbb{Z}^d$ with $k$ active particles at the origin and no particles elsewhere, and let $A_k$ be the aggregate, namely the set of sites visited a…

Let $X_e$ be independent and identically distributed random weights attached to the edges of a graph, such as the lattice $\mathbb{Z}^2$. For the associated first-passage percolati…

The KPZ fixed point is the random field … where $\gamma$ is the spatial coordinate and $\tau$ is the time coordinate. KPZ fixed-point universality conjecture. The KPZ fixed point i…

Uniqueness conjecture. In a thin enough linear wedge, the DLA cluster has only one infinite arm.

Let $f_t$ be the deterministic conformal map associated with the Loewner--Kufarev evolution at fixed time $t$, and let $z$ be a point in its domain. The cluster has half-plane capa…

The directed landscape $\mathcal L$ is the four-parameter scaling limit of Brownian last passage percolation: it encodes the last passage value from line $m$, location $x$ to line…

Let $\operatorname{SHL}(\alpha)$ denote the stationary Hastings–Levitov process obtained by normalizing slit sizes according to the conformal radius, for a parameter $\alpha\neq 0$…

In first-passage percolation, let $\xi(d)$ denote the wandering exponent governing the transverse fluctuations of time-minimizing paths in dimension $d$. Wandering-exponent conject…

Strict shape inclusion conjecture. If $p_1<p_2$, then

Shape-determined coexistence conjecture. For bounded initial sets, the possibility of coexistence is determined by the relation between the one-type shapes … .

KPZ exponent conjecture. It is conjectured that

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…

Remenik–Quastel conjecture. For KPZ growth models with an initial configuration that is flat with subdiffusive scaling, the limiting distribution is the same as for the correspondi…

KPZ class conjecture. Let $y$ be such that $\phi$ is twice differentiable at $y$ with $\phi^{\prime\prime}(y)\neq 0$, and set $\rho_0=\phi^{\prime}(y)$. If $0\leq\rho_0<1$,…