34 problems

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…

Fix $k$ and let $\mathcal{R}_k$ be the state space of the finite $k$-chain. For a partition $\lambda\in\mathcal{R}_k$, let $\lambda^{+i}$ denote the state obtained by adding a box…

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 $\mathbf{K}$ be a compact convex set of area $1$, let $\lambda>0$, and suppose that $\operatorname{argmax}\Phi_{\lambda,\mathbf{K}}$ consists of the single element $C^\star$, w…

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

Let $x,y\searrow 0$ and let $\beta=\beta(y)=-K/y$, with fixed $K>0$. Let $\mathfrak{W}(u\mid x,y,\beta(y))$ be the Grothendieck limit shape, and let $\Omega$ be the Vershik–Kerov–L…

Let $x,y\searrow 0$ and let $\beta=\beta(x)\nearrow 0$ satisfy $-\beta(x)\ll x$. Let $\mathfrak{W}(u\mid x,y,\beta(x))$ denote the Grothendieck limit shape, and let $\Omega(u)$ be…

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 $\Omega$ be the open diamond … and let $\Phi$ be the limiting variational function for a homogeneous Poisson point process on $\Omega$; its derivative $\Phi'$ is the correspond…

Let $\mu_{n,k}(\lambda\mid\{x_i\}_{i=1}^n,\{y_j\}_{j=1}^k)$ be the character measure on partitions $\lambda\subseteq k^n$ defined using Schur functions, and take…

Corner permuton limit conjecture. If $k,n\to\infty$ with $k/n\to\alpha\in\mathbb{R}_{>0}$, then

Circular limit-shape conjecture. As $n\to\infty$, the limit-shape curve is

Limiting density measure conjecture for minimal membranes. As $t\to\infty$, the measures $\mu_{L,t}$ converge to a limit measure

Limit shape conjecture for minimal membranes. There exists a unique surface $S\subset\mathbb R^n$ with boundary $\langle L\rangle$ such that, for every $\epsilon>0$, there is $N>0$…

Pathwise asymptotic enumeration conjecture. As $n_N\to\infty$ and $\varepsilon\to0$,

The variational principle for skew Young tableaux. The function $g(t,x)$ maximizes

Let $S=[0,1]^2$, and let $\phi^N$ be a stochastic interface model on $S_N$ with interaction potential $V=\mathsf{bead}$ or $V=\mathsf{\exp}$. Suppose its weight functions are … wher…

Let $G$ be a nested fractal graph, and let $o$ be a starting vertex such that every graph-metric ball $B_o(n)$ has spatial symmetry. Consider the four single-source Laplacian growt…

AHRV frozen-region conjecture. As $n\to\infty$, the boundary of the limit shape of $F_t$ is