63 problems

Kenyon–Okounkov conjecture. In the interior of $\mathcal{L}$, the fluctuations are governed by the pullback of the Gaussian free field from $\mathbb{H}$.

Kenyon–Okounkov conjecture. The limit of the height fluctuations of $h_{Omega_\delta}$ is the Gaussian Free Field in a non-trivial complex structure determined by the limit shape.

Differentiability and strict convexity conjecture. If the edge-weight distribution is not deterministic, then the limit shape is differentiable; if the distribution is continuous,…

Let $\rho$ be the uniform distribution on the diamond … Let $\Phi'$ denote the limit shape of the Young diagram associated with the corresponding locally uniform random permutation…

Eden model limit-shape conjecture. For every dimension $d\geq 2$, the limit shape of the Eden model is not a Euclidean ball.

Let $\mu_{m,n}$ be the Plancherel-Hecke probability measure on partitions $\lambda\subset\delta_n=(n,n-1,\ldots,1)$, defined by … where $d^{\lambda}(n)$ counts increasing tableaux…

Axial curvature and fluctuation exponent conjecture. For $r>0$,

Consider the northeast model on the integer lattice, with all sites in the first quadrant initially set to spin $0$ and all other sites initially assigned independently by $p$-coin…

Let $H^{\alpha}({\mathbf x},{\mathbf y})$ be the Busemann function in direction $e^{i\alpha}$, defined from the passage-time difference between the vertices associated with…

Limit surface conjecture. For each continual Young diagram $f$, there should exist a limit surface $L_f$ defined on $D_f$, bounded between the $x$-axis and the graph of $f$, that d…

Let $\alpha$ and $\beta$ be parameters satisfying … Let $t$ be the interface parameter, and let a smooth region mean a smooth region of the limit shape to the left of the interface…

Macroscopic boundary conjecture. The macroscopic boundary of the $t$-split two-periodic Aztec diamond to the left of the interface is governed by this saddle function: the boundary…

RBPD limit-shape and arctic-curve conjecture. As $n\to\infty$, the height function converges to a deterministic limit shape consisting of four frozen regions adjacent to the domain…

Let $\mathcal C_k$ denote the limit shape of the random-core growth process for fixed $k$. Let $D_k$ be the piecewise-linear curve with vertices … where $\gamma$ is the scaling con…

Let $H$ be a JUE random matrix with eigenvalues, and let $\widehat H$ be obtained by removing the last row and last column of $H$ and taking its eigenvalues. Construct a random pie…

Let $\mathbf{K}$ be a compact convex set in $\mathbb{R}^2$ with non-empty interior, let $\lambda>0$, and define … for compact convex sets $C\subseteq\mathbf{K}$. Write…

Let $(P_n)_{n\geq 0}$ be the suitably scaled maximal entropy random walk pyramidal process. Define … and let $\mathcal{H}$ be the graph of $\Omega$. Set … and…

Consider the specializations … Let $\mu_{n,k}^b$ be the corresponding measure, with $q>0$, $b\neq 0$, and let $\rho(t)$ and $\Omega(t)$ denote its limiting density and limit shape.…

Let $\mu_{n,k}^{\mathrm{Sp}}$ be the symplectic measure defined by the paper, with specializations … Let the associated symplectic Young diagram be sampled using the Berele samplin…

Consider the generalized principal specialization $x_i=q^{i-1}$ and $y_j=q^{b(j-1)}$, in the regime where the resulting limit shape is linear rather than flat at the corner. Let th…

Let $t_+$ be the right endpoint of the support of the limit shape, let $z_+$ be the corresponding root, and let $Omega$ denote the limit shape. Critical-corner classification conj…

For fixed $p,q\in(0,1)$, let $\boldsymbol{w}^{(q)}(n)\in S_n$ be the random permutation from the $q$-reduced pipe-dream model, and let $H^{(q)}$ be its height function. Let…

Let $\Phi(r)$ be the increasing function defined by the $L^1$ limit for maximal decreasing subsets of a homogeneous Poisson point process in the sloped rectangle: for $r\ge 0$, if…

Let $\omega(x)$ be the continuous limit shape of a sequence of Young diagrams, and let the associated random Young tableaux have a limiting surface. Continuity conjecture. If…

Let $\mathfrak{W}(u)=\mathfrak{W}(u\mid x,y,\beta)$ be the curve obtained from the surface and implicit-curve procedure described immediately beforehand, and consider Grothendieck…