15 problems

Bergman measure equilibrium conjecture. The normalized Bergman measure

Let $e_g(-s)$ be the generating function associated with fixed genus $g$, let $s$ be the parameter controlling the vertex valence, and let the equilibrium measure have support endp…

Let a ball carry an equilibrium measure in the external field considered in the cited work, and let its support be the set on which that equilibrium measure is concentrated. Shell-…

Let $d\geq1$ and $\max(0,d-2)<s<d$. Let $\overline{\mathbb B}$ be the closed unit ball, let $\mathbb S$ be its boundary sphere, and let $S_{Q,\overline{\mathbb B}}$ denote the supp…

Let $V$ be a radially symmetric potential such that the disk $\{z\in\mathbb{C}:|z|<R\}$ is contained in the droplet. Let $d\sigma_V=\Delta V\,dA=\rho\,dA$ be the equilibrium measur…

Moments of Riesz $p$-equilibrium measures are minimal for the ball. For every $q\in(0,\infty)$,

Euler-index conjecture. The precise asymptotic behaviours of the partition function depend on the Euler index of the droplets.

Radial arcsine conjecture. The radial arcsine distribution solves the full-space equilibrium problem in $\mathbb{R}^d$ for all dimensions $d$.

Consider the weakly admissible equilibrium problem in $\mathbb{R}^d$ with $d-2\leq s<d$ and an attractor–repellent external field generated by points $y_1=(0;y_{1,d+1})$ and…

Carrillo–Huang conjecture. These steady-state solutions should be the unique global minimizers of the associated interaction energy.

On $\mathbb{R}^d$, let $\mathrm{d}\mu(x)=\exp(-\|x\|^2)\,\mathrm{d}x$, and let $\mu^\ast$ be the probability measure defined as the weak limit of the rescaled polynomial-kernel mea…

For a time $t$, let $I(t)$ be the domain consisting of the allowed intervals $[a_k,b_k]$, and let $y_t(x)$ be the spectral curve governing the equilibrium measure, whose numerator…

Let $\tilde{\gamma}_j$ be neighborhoods of the connected components $\gamma_j$ of the support of the equilibrium measure, and set $\tilde{\Gamma}=\coprod_{j=1}^r\tilde{\gamma}_j$.…

One-cut regularity conjecture. Based on the analogy with matrix models without external source, the symmetry of the equilibrium measure, and numerical simulations, $V$ is one-cut r…

Let $K$ be a compact subset of a sphere $S(x_{0},r)$ with $C_{2}(K)>0$, and let $D$ and $\Omega$ be two Greenian domains containing $K$. Write $D^{*}$ for the inverse of $D$ with r…