12 problems

Airy edge conjecture. The edge limit of the additions should still be the Airy $beta$-process under suitable rescaling, even when some $kappa_l$ are negative.

Edge universality conjecture. We have

Let $d\geq 1$, let $\mathscr Q:\mathbb C^d\to\mathbb R$ be $C^2$ and strictly plurisubharmonic, and suppose that its droplet $S_{\mathscr Q}$ has smooth boundary. For each…

Let $d\geq 1$, let $\mathscr Q:\mathbb C^d\to\mathbb R$ be $C^2$ and strictly plurisubharmonic, and let $S_{\mathscr Q}$ be its droplet with smooth boundary. For…

Let $\lambda$ be a random Young diagram with respect to the skew Howe measure $\mu^{\mathrm{H}}$, and let $\lambda_i$ denote its row lengths. Let $t_+$ be the right edge parameter…

Airy edge universality conjecture. Under some mild conditions on the summands, the upper edge limit of $Vec{\lambda}(N)$ near $mu_+$ is still given by $\operatorname{Airy}(\beta)$…

GUE-corners universality conjecture. At such tangency points, the GUE-corners process is the universal scaling limit.

Universal edge fluctuation conjecture. The local statistics of uniform lozenge tilings for polygonal domains have a universal scaling limit at each particular type of points.

Let $N^{\varepsilon-1}\leqslant p\ll N^{-2/3}$, and consider a sparse random matrix in the regime described above. Suppose that the explicit random correction terms constructed to…

Edge-universality regime conjecture. The assertion in (1.2) holds if and only if

Consider the random normal matrix ensemble in a strip with the soft/hard edge condition, meaning that the gas is completely confined to the droplet $S_n$ by setting…

Let $A_n$ be a square random matrix with independent and identically distributed centered entries of variance $1/n$ and finite fourth moment, and let its spectral radius be the max…