65 problems

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.

Let $X$ be $d$-dimensional super-Brownian motion, with total occupation local time $L^x$, range $\mathcal{R}$, and boundary local time measure $\mathcal{L}$ supported on…

Selberg distribution conjecture. Its distribution has Mellin transform

Archimedes configurations. For all $t\in(0,1)$, the scaled configuration converges in distribution to the Archimedes measure in the vague topology for random Borel measures on…

Let $\mu_{\mathrm{ISE}}$ be the integrated superBrownian excursion measure on $\mathbb{R}$, and let $Y(\lambda)$ be the non-negative random variable whose law is specified by the l…

Transport-distance finiteness conjecture. If $\Phi$ is $(p+\vartheta)$-uniform for some $\vartheta>0$, then

Let the Dyson hierarchical lattice have critical dimension $2$, and let $\overline{W}$ be any inverse temperature. Consider the vertex-reinforced jump process (VRJP) on this lattic…

Reverse-time distribution-function conjecture. Then $\overleftarrow{F}^{(t)}\Rightarrow\overleftarrow{F}^{(\infty)}$, where the random mass of $\overleftarrow{F}^{(\infty)}$ is dis…

Complex-parameter Hermite zero distribution conjecture. For every complex $s\in\mathbb{C}$, the probability measures $\llbracket H_n^*(\cdot;s/n)\rrbracket_n$ converge to a probabi…

Let a strictly jammed configuration mean a collectively jammed configuration that disallows all uniform volume-decreasing strains of the system boundary, and let an infinite config…

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…

Fractional-moment multifractality conjecture. The same asymptotic extends to $h\in[0,1]$.

A Riesz gas of index $s$ in dimension $d$ is a particle system governed by Riesz interactions of index $s$. Dereudre–Vasseur's conjecture. The number rigidity of a Riesz gas of ind…

Boundary lower-bound conjecture. The boundary variant may provide an asymptotic lower bound for the rate of convergence of the occupation measure of a single Brownian path.

Convergence-rate conjecture. The rate of convergence should be of order

Random-measure convergence conjecture. The limiting joint random measure is the deterministic measure whose density is the limiting typical kernel.

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…

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…

Boundary-neighborhood approximation conjecture. For some constant $c'>0$,

Let $X$ be $d$-dimensional super-Brownian motion, let $L^x$ be its total occupation local time, and define the random measures $\mathcal{L}^{\lambda}$ on $\mathbb{R}^d$ by … Let…

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…

For $\beta > 0$, let $\mu^{\beta}$ be the random measure from the Main Theorem and define … The random measure $C'_0 \mu^{\beta}$ is strongly related to the KPZ dual measure of the…

Let $D$ be a planar domain, let $x_0\in D$, and let $\nu_\varepsilon^\gamma$ denote the regularised random measure associated with the local times of planar Brownian motion started…