18 problems

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…

Active clustering query complexity conjecture. The average number of membership queries required is $(1+o(1))\sum_{i=1}^k i p_i n$. This conjecture concerns the optimal average que…

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…

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…

Let $t$ be a time, let $\Xi_{\le t}$ be the set of particles arriving by time $t$, and let $A(t,\xi)$ denote the measurable region associated with $\xi\in\Xi_{\le t}$. The regions…

Macdonald law-of-large-numbers conjecture. With almost sure convergence,

Central limit conjecture. The infinite vector

Positive-Plancherel-parameter conjecture. The technical assumption $\gamma=0$ in the law of large numbers can be dropped: the same convergence should hold for every triplet…

Let $\lambda,\mu\vdash n$ be independently chosen random partitions of $n$, and let $\chi^\lambda[\mu]$ denote the value of the irreducible character indexed by $\lambda$ on the co…

Let $\lambda\vdash n$ be a uniformly random partition of $n$, and let $\mu$ be a uniformly random self-conjugate partition of $n$. Write $\widehat\mu$ for the principal-hook partit…

Let $P_n$ be the set of partitions of $n$, and let $\widehat\rho_k$ denote the principal-hook partition of the staircase partition $\rho_k=(k,k-1,\ldots,1)$, where $n=k(k+1)/2$. Ch…

The parameters $\omega_n$ define models of random partitions, and $\omega_n$ converges to $0$ at speed $1/\sqrt{n}$ as $n$ tends to infinity. The fluctuations of the partitions are d…

Under the preceding critical-temperature conjecture, let $\xi\in\Delta_1$ be the weak limit of the ordered normalized cycle-length vector. Poisson–Dirichlet law conjecture. The dis…

Consider the cycle or loop model on the cubic lattice $\Lambda_n=\{1,\dots,n\}^d$ in $\mathbb{Z}^d$. Let $\lambda^{(i)}$ be the length of the $i$th-longest cycle, and let…

Let $(\lambda^{(1)},\lambda^{(2)})$ be a random bipartition under the B-$q$-Plancherel measure, and let $\lambda=\lambda^{(1)}\sqcup\lambda^{(2)}$. Asymptotic part-allocation conje…

Let $rad_1<1$ in the setting of Theorem, and suppose that $f$ has a pole at $rad_1$. Let $mu^{(n)}$ denote the corresponding measures and let $\alpha^{(n)}$ be their scaling functio…