15 problems

Let $A_k$, for $k\geq 0$, be independent $n\times n$ matrices whose entries are independent standard complex Gaussian random variables. The zeros in the unit disk of the determinan…

Let $f$ and $g$ be measurable functions defining stationary determinantal probability measures $\\P^f$ and $\\P^g$, and let $H(\\P^f)$ denote the entropy of $\\P^f$. Entropy concav…

Let $\mathcal{K}^{\mathrm{barcode}}_{(M)}(0,t)$ denote the finite-truncation barcode kernel, and let $t\in\mathbb{Z}_{>0}$. Exponential correlation-decay conjecture. The correlatio…

Let $q$ and $\kappa$ be the model parameters, let $\mathcal{F}_n^s(0)$ be the orthogonal functions appearing in the barcode kernel, and let $\mathcal{K}^{\mathrm{barcode}}(s,t)$ be…

Let $K^{\mathrm{barcode}}_{(L)}(s,t)$ be the pre-limit barcode kernel for $s,t\in\mathbb{Z}$, and let $q$ be the model parameter. Barcode-kernel limit conjecture. For all…

For $t\in\mathbb{Z}$ and $M\in\mathbb{Z}$, define the two truncation limits $\rho_0(t)$ and $\rho_1(t)$ by the displayed sums over $n\in\frac12\mathbb{Z}_{\ge0}$. Let…

Let $(\mathsf{t},\mathsf{x})$ be a macroscopic point in the waterfall region $\mathcal{W}$, and let $q$ and $\kappa$ be the model parameters. Let…

Universal Edge Fluctuation Conjecture. Under suitable assumptions on the convergence of $G_n$, all scaling limits of the determinantal point processes at the frozen boundary are gi…

Let $f\colon \mathbb{T}\to[0,1]$ be a Borel function, and let $\mu_f^{+}$ be the associated stationary determinantal measure on the metric space $(\{0,1\}^{\mathbb{N}},d)$, where ……

Let $\Phi_-(\tilde{s})$ be the large-deviation rate function for the left tail of the one-dimensional KPZ equation, and let the cumulant expansion of the determinantal-process repr…

Lyons–Peres conjecture. For $\mathbb{P}_K$-almost every $X\in\operatorname{Conf}(E)$,

Determinantal representation conjecture. The unique smooth normalized solution $\varphi$ may be represented as a limit in $L^{1}(X)$:

Let $r>0$, let $\tau=it_*$ with $t_*>0$, and let $\mathfrak{M}$ be the configuration space on $[0,2\pi r)$. For an initial configuration $\xi\in\mathfrak{M}$ with … and with … writ…

Let $cmathbb{S}^d$ be the unit sphere and let $n=cpi_L$. Let $K$ be an isotropic kernel producing $n$ points through its determinantal point process, and let $E_s(x)$ denote the Ri…

Let $\underline K\,_{z,z'}(x,y)$ be the Gamma kernel on $\mathbb Z'\times\mathbb Z'$, where $\mathbb Z'=\mathbb Z+\frac12$, and let Christoffel--Darboux kernels arise from generic…