36 problems

Baik–Deift–Johansson conjecture. As $n\to\infty$, the joint distribution of the first few rows of $\lambda$, after the proper shift and rescaling, is the same as the distribution o…

A partition is an $s$-core partition if none of its hook lengths is divisible by $s$; an $(s_1,s_2,\cdots,s_m)$-core partition is simultaneously an $s_i$-core partition for every…

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…

Baik–Deift–Johansson conjecture. The limit distribution of the $x_i$'s exists as $n\to\infty$ and coincides with the limit distribution of the $y_i$'s.

Let $\alpha\geq 1$ and let … For every real $x_0$, define the approximation error by comparing the distribution of $W_{\alpha}$ under Jack measure with the standard normal distribu…

Let $z,z'$ be parameters for which the gamma kernel $\unK^\gam(x,y\mid z,z')$ on $\Z'\times\Z'$ is defined as in Section 2. The paper proves under the restriction $|z+z'|<1$ that t…

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…

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 $(\sigma_n)_{n\geq 1}$ be conjugacy-invariant random permutations of size $n$. For a partition $\lambda$, let $L_\lambda$ denote its Russian-coordinate profile, let…

Let $(\sigma_n)_{n\geq 1}$ be conjugacy-invariant random permutations of size $n$. Suppose that … and … Writing $\lambda(\sigma_n)$ for the partition associated with $\sigma_n$ and…

Let $c$ be a fitness distribution, and consider the periods during which the largest table remains the same as a function of time. Ageing conjecture. For all fitness distributions…

Let $\nu\ge 3$, let $M=M(n)\to\infty$ with $M=\exp(O(n))$, and let $X_1,\dots,X_n$ be independent random variables uniformly distributed on $[M]:=\{1,\dots,M\}$. An ordered partiti…

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…

Condensate-density conjecture. The quantity

PMF condensate-density conjecture. The condensate density is

Mod-Gaussian fluctuation conjecture. This singular point corresponds to mod-Gaussian fluctuations, but with parameters $N_{n,k}$ and $D_{n,k}$ different from those in the generic c…

Rank conjecture. We conjecture that

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