17 problems

In cylindrical Hastings–Levitov$(0)$, let $b[3m\omega_{N, \lambda}b[0m$ be the last time a new tree grows on $b[3m\mathbb T_Nb[0m$, where $N$ is the cylinder parameter and…

Consider a hole whose boundary is a polygon with three sides and whose angles are integer multiples of $36^{\circ}$. The angle-sum condition permits two angle multisets,…

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…

Physical-origin conjecture. The physical origin of the intrinsic roughness is related to the uncertainty in the measurement of the isochrone arrival time (IAS).

The stationary Hastings–Levitov process with parameter $\alpha$, denoted $\operatorname{SHL}(\alpha)$, is the stationary version of the Hastings–Levitov model on the upper half-pla…

Periodic KPZ interpolation conjecture. The distribution function $F^{\mathrm{pKPZ}}_m$ interpolates the KPZ fixed point and one-dimensional Brownian motion.

KPZ universality conjecture. For a large class of such models, $\mathbf{h}_T(\gamma,\tau)$ converges, up to scaling factors, to a model-independent universal two-dimensional random…

Particle jump-and-coalescence conjecture. The resulting pictures converge to a meaningful particle jump-and-coalescence process.

First-passage percolation KPZ exponent conjecture. The exponents satisfy

Consider directed polymers in random environment in the strong-disorder phase. In the zero-temperature limit, directed polymers are related to last-passage percolation, whose integ…

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…

Let $F^0_{\rm stat}(r)$ denote the limiting distribution function for the KPZ equation with stationary Brownian initial data, and let $F_{\rm GOE}(r)$ denote the GOE Tracy–Widom di…

Let $\bar{B}(t)$ be the infected set at time $t$, let $T(\mathbf{0},\mathbf{n})$ be the hitting time from the origin to $\mathbf{n}$, and consider the location where $\bar{B}(t)$ h…

KPZ class conjecture. Suppose that $\phi$ is twice differentiable at $y$ with $\phi”(y)\neq0$ and set $\varrho=\phi'(y)$ for $|\varrho|<1$. If $A(\varrho)<\infty$ and…