90 problems

Johansson's conjecture. Almost surely, $x\mapsto\mathcal{A}(x)-x^2$ attains its maximum at a unique point $Y\in\mathbb{R}$.

Stationary-measure conjecture. The process $(\widetilde H_t)_{t\in[0,\tau]}$ represents the stationary measure of the KPZ equation on the interval $[0,\tau]$ with Neumann boundary…

Let $rho ameq{}(0,1)$ be the stationary Bernoulli density, let $mathbb{P}rho$ be the corresponding TASEP path measure, and let $ht(j)$ be the height function. For $winmathbb{R}$, d…

Let $(\Omega,h)$ be a quantum disc as defined in the cited work, and let $\mathbf{K}(t)$ denote the corresponding heat-content quantity. Quantum-disc heat-content conjecture. As…

Let $(\Omega,h)$ be a quantum disc as above, and let $\mathbf{H}(t)$ denote its heat trace, with $c_\gamma$ the leading Weyl-law constant and $\mu(\Omega)$ its quantum area. Quantu…

Let $\mathfrak{h}$ be an upper-semicontinuous function that is bounded above by a linear function, and let $\mathfrak{h}^{\mathrm{cutoff}}_\mathrm{p}$ be its periodized cutoff. Let…

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…

Airy-type characterization conjecture. Then $\mathcal{L}$ is equal in distribution to the $\mathrm{KPZ}_t$ line ensemble, up to an independent affine shift.

Parabolic rigidity conjecture. As $n\to\infty$, the sequence of functions

Asymptotic gap conjecture. Almost surely as $n\to\infty$,

Let a one-dimensional microscopic interface growth model belong to a broad class of such models, and let $\mathcal H$ denote its macroscopic fluctuation field. The KPZ equation is…

KPZ fluctuation conjecture. At the critical scaling, one-dimensional diffusions in random media should exhibit fluctuations governed by the KPZ equation.

Directed landscape conjecture. For even larger scales

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…

Critical KPZ scaling-limit conjecture. There exists a non-random function $b(\varepsilon)>0$ with $b(\varepsilon)\to\infty$ as $\varepsilon\to0$ such that, for every…

Uniform-path comparison conjecture. For all $t e(0,1/2]$,

The KPZ line ensemble is a collection of random curves associated with the KPZ equation, and the fundamental solution is the solution kernel that propagates initial data for that e…

Let $K_n(f)$ be the spatially averaged fluctuation observable, written as a sum of identically distributed terms $K_n(x)$, and suppose these terms possess all moments. Gaussian sca…

Product-Bernoulli determination conjecture. For every $\alpha\in[1,3/2]$, the renormalization constant $\mathscr{R}_{N}$ is still determined by product Bernoulli measures.

KPZ universality conjecture. The Kardar–Parisi–Zhang equation should be a universal scaling limit for stochastic growth fluctuations, including phenomena such as burning fronts, ba…

Let $N$ be fixed, let $\gamma$ denote the parameters of the coupled Markov chain, and consider the coupled chain defined by, its dual chain defined by, the system of SDEs, and its…

Let $\Sigma_L(\beta)$ denote the effective diffusivity of the winding number of a directed polymer in a white noise environment with period $L$, and let $\Sigma_1(\beta)$ be the di…

For $theta inmathbf{R}$, let $mutheta$ be the law of two-sided Brownian motion with drift $theta$, viewed as an invariant measure for the recentered KPZ process $h(t,x)-h(t,0)$. **…

Let $\mu\in\mathscr{N}_+$ and write $v:=\partial_x g:=\partial_x\mathsf{H}[\mu]$, where $v$ is a tractable solution. Non-entropy shock comparison conjecture. The measure $\mu$ rest…

Let $h=h(x,t)$ solve the two-dimensional anisotropic KPZ equation … where $Q$ is a symmetric diagonal $2\times2$ matrix, $\xi$ is space-time white noise, and…