26 problems

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…

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 $n\to\infty$,

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…

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 $\mu\in\mathscr{N}_+$ 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…

Theorem 1.1 limit-shape conjecture. The results of Theorem 1.1 hold for all $\vec{\mathbf{r}}\in\mathscr{R}$ under $N\geq1$ and $N^2T\to\infty$.

Let $T_0$ and $T_t$ be the passage times at perturbation times $0$ and $t$, and write $\operatorname{Var}(T)$ for the variance of the passage time and…

Let $\mathfrak S_n$ be a uniform meandric system of size $n$, and order its loops by decreasing size. For each fixed $k\in\mathbb N$, let the $k$th largest loop be measured by its…

Geodesic--argmax rigidity conjecture. For every $\tau\in(0,1)$ and $T>0$, conditionally given $\mathcal L(0,0;0,T)=L$, as $L\to\infty$,

Let $\mathcal{H}_{u,v}$ denote the explicitly constructed stationary processes for the half-space KPZ equation with Neumann boundary parameter $u$, indexed by…

Non-uniqueness-times conjecture. For nice enough initial conditions $h_0$, almost surely $\mathcal{T}_{\geq2}(h_0)$ is dense in $[0,\infty)$, $\mathcal{T}_{\geq5}(h_0)$ is empty, a…

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

Fix $t>0$, $\Delta>0$, two segments $[X,X']$ and $[Y,Y']$ in $\mathbb R$, and real numbers $(x_1,\dots,x_n)$, $(y_1,\dots,y_n)$, $(\tilde x_1,\dots,\tilde x_k)$, and…

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define … Here $b$ is a standard Brownian bridge whose average is represented by…

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define … Here $\max$ denotes the maximum over the bridge parameter. Let…

Let $n$ be a non-negative integer and let $h_0$ be an arbitrary sufficiently regular continuous function satisfying … For independent standard Brownian bridges $b_1,\ldots,b_n$, de…

Let $R$ be the renormalisation transformation on point fields of concentration points and shocks, and suppose the density decays as $n^{-\alpha}$ for a critical exponent…

Let the Hamiltonian be convex and let its associated Lagrangian have superlinear growth. A generalized directed polymer is the polymer process associated with the corresponding ran…

In one spatial dimension, let $\Phi$ be the global solution with $a=b=0$, and regard it modulo additive constants so that it has stationary spatial increments. Brownian limit conje…

Assume the hypotheses of the one-force-one-solution conjecture, and work almost surely. A one-sided minimiser is a curve defined on $(-\infty,t]$ such that compact perturbations fi…

Let $d\in\mathbb{N}$ and let $F_\omega(t,x)$ have exponential decay of space-time correlations. For every $b\in\mathbb{R}^d$, consider global solutions of the random Hamilton–Jacob…