162 problems

Let the KPZ equation be the stochastic growth equation whose solution is denoted by $\mathcal{H}_\beta(T,X)$, and let the KPZ universality class consist of models sharing its chara…

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…

Quadratic mixing-time conjecture. The correct order of the mixing time in Theorem 2 is $N^2$.

KPZ universality conjecture. All KPZ models, under the so-called $1:2:3$ scaling, converge to the KPZ fixed point.

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…

Superdiffusive KPZ scaling conjecture. The KPZ scaling profiles, and hence superdiffusive dynamics characterized by the anomalous dynamical exponent $z=3/2$, occur across all such…

Let an Airy line ensemble be a non-intersecting line ensemble with the Brownian Gibbs property, and suppose that after subtracting a parabola its law is stationary. A height shift…

In kinetic roughening, let $C_\mathrm{t}(t_1,t_2)$ be the connected two-time correlation function and suppose its rescaled form has the long-time-ratio decay…

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…

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

The Airy$_1$ process is the limiting process governing fluctuations in certain growth models. Let the GOE Dyson Brownian motion be the matrix-valued stochastic process obtained by…

A one-dimensional stochastic system has local interactions, a conserved density or height-gradient field, and a weak asymmetric perturbation of a reversible dynamics. Let…

A half-space Gibbsian line ensemble is expected to have a scaling limit called the pinned half-space Airy line ensemble. Locally, the curves are ordered line ensembles with pinning…

Let $d=2$, let $X_n$ be a random walk in an i.i.d. uniformly elliptic random environment, and let $I_q:\mathbb R^2\to[0,\infty]$ be its quenched large deviation rate function. Writ…

Let $X_n$ be a random walk in an i.i.d. uniformly elliptic random environment on $\mathbb Z^2$, and let $P_{0,\omega}$ denote its quenched law. Under the specific scaling appropria…

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…

Logarithmic-shift BK conjecture. The KPZ line ensemble satisfies a form of the BK inequality with a shift that is logarithmic in the size of the interval over which the first curve…

The directed landscape $\mathcal L$ is a four-parameter scaling-limit object constructed from Brownian last passage percolation. Universality conjecture. It is conjectured to be th…

The KPZ universality class contains many stochastic growth models with universal spatial and temporal fluctuations. The Airy process and related Airy line ensembles describe spatia…

The parabolic Airy process is … where $\mathcal{A}_2$ is the Airy$_2$ process. Universality conjecture. The parabolic Airy process is conjectured to be a universal limit of models…

Directed landscape universality conjecture. The directed landscape is conjectured to be the universal scaling limit for various models in the KPZ universality class.

A bi-infinite geodesic is a geodesic path extending infinitely in both directions in a model of the KPZ universality class. Nonexistence conjecture. With probability one, bi-infini…

Consider the infinite Brownian particle systems described by the order-statistics formulation, with particles whose asymmetry parameter satisfies $p\neq 1/2$, and their connections…

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 $(0,1)$.…

Let $\mathcal{H}^{\mathrm{UT}}$ and $\mathcal{H}^{\mathrm{UT}}_0$ be the upper tail fields of the KPZ fixed point. Upper-tail-field initial-condition independence conjecture. The f…