17 problems

Let $\mathbb W$ denote the renormalized Coulomb energy of unit-density point configurations in the plane. The triangular lattice conjecture. The triangular lattice is a global mini…

Rosenzweig's optimal scaling conjecture. This scaling assumption should be in general optimal: the incompressible Euler equation should not be the limiting evolution of the empiric…

Let $d\geq 2$ and let $\beta>0$. Consider a translation-invariant Coulomb interacting Brownian motion in $d$ dimensions, and denote its self-diffusion coefficient by…

Consider the variance of linear statistics with respect to $mu_n$ for the three methods listed in the source, including MCMC and Gibbs-measure backgrounds. Gibbs-measure improvemen…

The experiments compare quenched Gibbs measures using MCMC backgrounds with backgrounds obtained by reweighting a realization of a Coulomb gas, where $n$ points are used to approxi…

For a two-dimensional Coulomb gas with quadratic potential $Q(z)=|z|^2$ and general $\beta>0$, let the large-$N$ expansion in question be the expansion of the logarithm of the scal…

Let $V$ be a regular potential independent of $N$, and let $\sigma_V$ be its associated equilibrium measure and $I_V[\sigma_V]$ the corresponding electrostatic energy. Assume that…

Consider the random points $\mathrm{z}_1,\ldots,\mathrm{z}_n\in\mathbb{C}$ from the point process discussed in the source, with inverse-temperature parameter $\beta$ and limiting e…

Let $\mu$ be a probability measure and let $S:=\operatorname{supp}\mu$. Assume that $S$ is compact, $U$ satisfies Assumption, $\partial U$ is piecewise smooth, and $U\subset S$. Su…

In the two-dimensional setting, let $\beta_p$ denote the perfect freezing transition and let $n$ be the number of particles. The notation $\beta_p\lesssim\log n$ means that…

Let $V$ be strongly confining and let $X_{n,1},\ldots,X_{n,n}$ be the Coulomb-gas points, with empirical measure $\mu_n=n^{-1}\sum_{k=1}^n\delta_{X_{n,k}}$. For a smooth enough tes…

Let $N$ particles form a Coulomb gas at inverse temperature $\beta$, and consider observation lengthscales compared with the threshold $\beta^{-\frac12}N^{-\frac1{\mathsf d}}$. A p…

The JLM law. For $R\to\infty$, after taking the number of particles $N$ to infinity, the probability of this event satisfies

Collapse conjecture. As $c$ approaches its critical negative value $-2$, the ensemble collapses at the origin, meaning that the one-particle density converges to a Dirac delta at t…

In the three-dimensional Coulomb gas and log-gas with Euclidean confinement, let the edge refer to the outer boundary of the equilibrium particle distribution, and consider the flu…

Perfect-lattice crystallization conjecture. The minimizer of the renormalized energy $\mathcal{W}$ is a perfect lattice.

Askey–Wimp–Kerov stationary-density conjecture. The stationary density for a large system is again given by the Askey–Wimp–Kerov distribution $\rho_c$ for the parameter range