19 problems

Let $h(x,t)$ be the height function of the polynuclear growth (PNG) model on a one-dimensional substrate with flat initial conditions. Let ${\cal A}_{\rm 1}$ denote the Airy one pr…

Consider planar last-passage percolation with geometric weights. Let $T_n(\omega)$ and $T_n(\tilde\omega^t)$ denote the last-passage times in the original and site-resampled enviro…

Let the KPZ fixed point have a general initial condition that grows sufficiently more slowly than $f(x)=x^2$ as $|x|$ becomes large, so that the KPZ fixed point exists at $(0,1)$.…

Universal backward-geodesic endpoint conjecture. There exists a universal constant $c_2$ such that

Universal Airy-one scaling conjecture. There exist universal constants $c_1$ and $c_2$ such that

Let $D_N(t)$ be the discrepancy between the backward-geodesic endpoint index $N(t\downarrow0)$ and the corresponding minimization index in the exclusion-process setting considered…

ASEPsc particle-position conjecture. For any $N\in\mathbb{Z}$ and every $s\in\mathbb{R}$,

ASEP backward-geodesic endpoint conjecture. For any $N\in\mathbb{Z}$,

Let $\mathcal{Z}(s,x;t,y)$ denote the continuum directed random polymer partition function, and define … Let $\mathcal{S}(x,y)$ be the parabolic Airy sheet. KPZ sheet-to-Airy-sheet…

A deposition process is a mixture of random deposition (RD) and ballistic deposition (BD), with a fixed positive amount of BD, and its block heights have light-tailed distributions…

Tracy–Widom GUE fluctuation conjecture. One has

Consider a particle-conserving system on a ring of size $N$ with only nice local interactions, and let the average current be a function of the particle density. Assume that the se…

Let $X$ be the set of triples $(y,s,t)$ with $y\in\mathbb{R}$ and $s<t$. Equip $X$ with the metric … For $k\geq2$, let $E_k$ be the subset of triples with $|s-t|\geq1$ for which th…

Fix $\theta,r>0$ and let $\tilde{f}_N^{LG}$ be the random continuous function from Definition 2. Then, as $N\to\infty$, the random functions $\tilde{f}_N^{LG}$ converge weakly in…

Consider the fractional stochastic Burgers equation (ec1.5.3). Let $\mathcal{Y}_t^{m,\rho}$ denote a stationary energy solution when it is unique, and let $\mathcal{X}_t^\rho$ be t…

Let $P$ be the pressure function, let $\beta$ be the slope of flat initial conditions, and set $\lambda=\gamma P”(\beta)\neq 0$ and $A=-P'(\beta)$. For the particle system with fla…

Uniqueness conjecture. For every $c\in\mathbb{R}$, there is a unique Brownian Gibbs line ensemble $\mathcal{L}\in\mathscr{G}_{\Theta}$ such that

Sheffield's extremal Brownian Gibbs line ensemble conjecture. As a convex set, the extremal points of $\mathscr{G}_{\Theta}$ are exactly $\{\mathcal{L}^c\mid c\in\mathbb{R}\}$.

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…