14 problems

Let $\alpha(n)$ be the random completely multiplicative function appearing in the paper, let $\mu$ be the Möbius function, and let $m_{x,\infty}$ be the random measure defined in t…

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…

Let $\mathbb L_n$ be the set of vertices at level $n$ of the rooted regular tree with branching factor $b$, let $L_t(x)$ denote the local time at vertex $x$ up to time $t$ for simp…

Let $(X(k))_{k \geqslant 1}$ be a sequence of independent random variables uniform on $\{\pm 1\}$, and define $(A(N))_{N \geqslant 0}$ by the paper's formal power series. For any f…

Let $(A(N))_{N \geqslant 0}$ be defined by the same formal power series from a sequence $(X(k))_{k \geqslant 1}$. Real-Gaussian and Rademacher extension conjecture. The first-momen…

Let $(A(N))_{N \geqslant 0}$ be defined by the formal power series associated with a sequence $(X(k))_{k \geqslant 1}$ of independent standard complex Gaussians. First-moment asymp…

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.

Let $\beta_c=2$ and consider the random densities from the preceding conjecture, with the critical exponent $\beta=\beta_c$. Critical multiplicative-chaos conjecture. The preceding…

Let $T\to\fty$, let $\beta\in(0,2)$, and consider the random densities on $[0,1]$ given by … Multiplicative-chaos conjecture. These random densities should converge in distribution…

Let $X$ be the normalized exponential functional introduced in the source, and let $q$ be in the range where the displayed moments exist. Freezing hypothesis. For $\beta>1$, … This…

Let $V_N$ be the maximum of the discretized Gaussian free field on the circle with singularity parameter $\alpha\geq 0$. Let $M_{(\tau=1,\alpha,\alpha)}$ denote the critical Morris…

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…

Let $M^{\gamma}$ be the subcritical chaos measures for $\gamma^2<2d$, and let $M^c$ be the positive random measure obtained in Proposition 3.1 as a limit…