59 problems

Let $\mathsf{L}_N$ be the C$\beta$E field on $\mathbb{T}$, with parameter $\beta>0$, and suppose the critical relation $\beta=2s^2$ holds. Let $\mathfrak{G}_\beta$ denote the expli…

Let $\gamma=\alpha+\mathrm i\beta\in\mathcal D$, where $\mathcal D$ is the domain of parameters for which the Gaussian multiplicative chaos measures $\mathrm M_{\gamma}$ and…

Morris distribution conjecture.

The Gaussian multiplicative chaos (GMC) is a random fractal measure obtained from a log-correlated Gaussian field $X$ on a domain $D\subset\mathbb{R}^d$ by the renormalized density…

Let $\mu_\gamma$ be Gaussian multiplicative chaos on the unit circle with parameter $\gamma$, and define the Fourier dimension $\operatorname{dim}_F(\mu)$ as the largest $s$ such t…

Selberg distribution conjecture. Its distribution has Mellin transform

Moment-bound conjecture. The same bound holds for all $2\leq p<\infty$.

Let $\gamma\in(1/\sqrt{2},\sqrt{2})$, let $\mu_\gamma$ be the Gaussian multiplicative chaos measure on $[0,1]$ with parameter $\gamma$, and let $c_n$ be its $n$-th Fourier coeffici…

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…

Optimality conjecture. The sufficient condition

Let $M^{\theta}$ denote the polymer measure associated with the stochastic heat flow at parameter $\theta$. A conditional Gaussian multiplicative chaos conjecture. If one takes…

Small-deviation conjecture. The limit

Let a random planar map be conformally embedded with its three roots sent to fixed points $\mathbf{z}=(z_1,z_2,z_3)$ of the Riemann sphere. Let $\mu_n$ be the uniform measure on th…

Beltrami-welding conjecture. The above Beltrami welding yields an SLE loop measure and thus an independent second construction of the SLE loop in Theorem 1.3 of Ang (2021).

Let $Q(a,a+x)$ denote the inverse Gaussian multiplicative chaos increment over the interval $[a,a+x]$, and let $Q(x)$ denote the corresponding unshifted inverse. Shifted inverse-mo…

Let $d\in\mathbb{N}$ and let $\gamma>0$. Define the free-energy function $\lambda(\gamma)$ by … Set … and let $\gamma_c$ denote the critical parameter introduced earlier in the pap…

GMC universality conjecture. The result should correctly describe the behaviour of analogous moments of moments of random matrices, or more generally of structures which are asympt…

Limiting-variance conjecture. One should have

Let $\textnormal{MoM}_N^{(\beta)}(k;q)$ denote the $k$-th moment of the partition-function moments for the circular beta ensemble. At moment-criticality … with $k>1$ and with…

Let $\gamma$ belong to the glassy phase ${{\mathcal P}} _{\mathrm {II}}$, and let $M^{(\gamma)}_\varepsilon$ be the regularized complex Gaussian multiplicative chaos distribution a…

Three-point GMC law conjecture. The law of GMC on the Riemann sphere with three insertion points $(\alpha_1,\alpha_2,\alpha_3)$ is conjectured to be given by a product of independe…

The conjectural three-point GMC law.

Let $\zeta(s)$ denote the Riemann zeta function, let $T\to\infty$, and let $\omega$ be as in the zeta-function model described in the source. For $\beta<2$, consider the subcritica…

The sine-process Gaussian multiplicative chaos convergence conjecture. The measure $\mu_{T,A,p}$ converges, as $T\to\infty$, in law in the space of Radon measures on $[0,1]$ endowe…

Let $D\subset\mathbb{R}^2$ be a bounded domain, and let $\mu$ be the random measure appearing in the limiting extremal process of the discrete Gaussian free field, characterised by…