10 problems

Let $D\subset\mathbb{R}^d$ and let a log-correlated Gaussian field on $D$ admit a smooth white noise decomposition. Denote its derivative Gaussian multiplicative chaos by…

Let $U_N$ be a random matrix distributed according to the Haar measure on the group of $N\times N$ unitary matrices, and let $p_N(\theta)=\det(1-U_N e^{-i\theta})$ be its character…

For $x\in\mathbb{Z}^2$ and $N\geq 1$, let $\ell_x^N$ be the total number of visits to $x$ by a planar simple random walk started at the origin before exiting $[-N,N]^2$. Define the…

Fyodorov–Keating conjecture.

Universal maximum conjecture. The maximum satisfies, for some constant $C$ and a random variable $Z$ called the derivative martingale,

Triple-point conjecture. As $\varepsilon\to0$,

Supercritical renormalization conjecture. As $\varepsilon\to0$,

Let $(X_n)_n$ be a cutoff approximation of a centered Gaussian field on $[0,1]^d$ with logarithmic covariance, and set $c_n={\rm Var}(X_n)$. Let $M'$ be the derivative chaos and le…

Maximum-law conjecture. There is a constant $c>0$ and a limiting random variable $G_d$ such that $Y_t$ converges in law to $G_d$ as $t\to\infty$, with

For $\gamma^2>2d$, let … set $\alpha=\frac{\sqrt{2d}}{\gamma}\in(0,1)$, and let $M'$ be the derivative-martingale measure. Let $N_\alpha$ be an independently scattered random measu…