26 problems

The Menshikov–Petritis conjecture. The additional conditions needed for the chaos problem to have nontrivial solutions should be the same as those needed for the random walk to be…

Let $\omega$ be the random shift appearing in the paper and define the random function or distribution … A real multiplicative chaos is a non-trivial random measure obtained from s…

Let $M_\gamma$ be the standard sub-critical Gaussian multiplicative chaos measure on the unit circle, with parameter $0<\gamma<1/\sqrt{2}$, and let $\dim_F(M_\gamma)$ denote its Fo…

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 $(X(k))_{k \geqslant 1}$ 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…

Let $T$ tend to infinity and consider the values of the Riemann zeta function on unit intervals along the critical line. Averaged local moment conjecture. … This conjecture is clos…

Let $l_x^N$ be the total local time at $x$ for a simple random walk on $\mathbb{Z}^2$ started at the origin and stopped when it leaves $[-N,N]^2\cap\mathbb{Z}^2$, and let $\nu'$ de…

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…

Let $d$ be the dimension, let $\gamma=\alpha+i\beta$, and define the phase-II parameter region by … This region is called the glassy phase. Phase-II conjecture. In this regime, it…

Let $M^{(\gamma)}_\varepsilon$ denote the regularized complex Gaussian multiplicative chaos associated with the complex parameter $\gamma$. The parameter $\gamma$ lies outside the…

DOZZ moment conjecture. The moment is

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 $M_\beta$ be the multiplicative-chaos limit random measure on the circle, and let $M_{(\tau,\lambda_1,\lambda_2)}$ be the Morris integral distribution. Restrict to … with…

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…

Law of total mass. The generalized total mass satisfies

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…

Let $\zeta$ denote the Riemann zeta function, and consider its logarithm on the critical line, at points far away from the origin and on a suitable scale. The log-correlated Gaussi…

Let $D$ be a domain and consider the discrete Gaussian free field in $D$ on a lattice with mesh $\varepsilon$, with $\varepsilon\to0$. Let $M'(D)$ denote the total mass of the deri…

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…

Let $M$ be a lognormal star-scale invariant random Radon measure, with Gaussian factor $\omega_\varepsilon(r)$ in the star-equation. Moment-scaling conjecture. There exists…

Let $M$ be a random Radon measure satisfying lognormal star-scale invariance: for every $0<\varepsilon\leqslant 1$, there are a stationary stochastically continuous Gaussian proces…