19 problems

Let $\mathcal{J}^{(m,n)}_{\alpha}$ be the Coulomb gas integral associated with a $(n,m)$ chordal link pattern $\alpha$, where $m\leq (n+2)/2$. These functions satisfy the null vect…

Let $\mathcal{K}^{(m,n)}_{\alpha}$ denote the screening solution constructed by Coulomb gas integrals from a chordal link pattern $\alpha$, in the presence of a marked boundary poi…

Let $\mathcal{J}(\boldsymbol{x},u)$ be a screening solution obtained by integrating the Coulomb gas integrand over an admissible choice of screening contours. These functions satis…

Let $\beta$ be the inverse temperature, with $4\pi\leq\beta\leq 8\pi$, and let $n$ be the integer part of $1/(2-\beta/4\pi)$. A neutral multipole is a multipole whose total charge…

Let $\operatorname{CLE}_\kappa$ be a conformal loop ensemble and consider general multichordal configurations with prescribed boundary connectivity data. Coulomb gas integrals are…

Consider the partition function of the two-dimensional Coulomb gas in the critical regime, whose expansion has been obtained with an error term of order $o(n^{1/4})$. Partition-fun…

Let a two-dimensional Coulomb gas develop a gap along a sufficiently smooth curve, rather than necessarily along a circle, and consider the new limiting kernel governing local corr…

Fix irrational $\kappa\in(0,8)$ and let $\mathrm{LP}(n,m)$ be the set of link patterns. Let $\mathcal{Z}_{\alpha}$ be the pure partition function indexed by…

Chordal ground-solution completeness conjecture. The chordal ground solutions $\mathcal{J}^{(m,n)}_{\alpha}$ constructed from chordal link patterns are exactly all solutions to the…

Fekete–Pommerenke energy conjecture.

Let $\mu_0$ be the unique minimiser over probability measures on $\mathbb R^2$ of … The density obtained formally from the Euler–Lagrange equation is $1/\pi$ on the support of…

Let $R_n$ be the one-point function of a two-dimensional Coulomb gas, and let $S$ be a general droplet with real analytic boundary. For a point $p\in\partial S$, write $\kappa(p)$…

Let $\hat\beta_{\mathrm{eff}}$ be the effective inverse temperature defined in the source, and let $f^{\operatorname{Coul}}(\beta)$, $f^{\operatorname{IV}}(\beta^{-1})$,…

Sharp free-energy exponent conjecture. The exponent $\pi^2\beta$ in this lower bound cannot be improved. Equivalently, the corresponding free-energy corrections should have exponen…

Let $(\beta_n)$ be a sequence of inverse temperatures, and consider the Coulomb gas configurations at these temperatures. Sharpness conjecture. If almost sure uniform separation ho…

Strong beta-ensemble expansion conjecture.

Mabuchi energy expansion conjecture. Under these assumptions,

Let $P_N(s)$ denote the distribution of the spectral linear statistic $s$, let $\rho_*$ be the equilibrium density associated with the constrained problem at $s$, let $S[\rho_*]$ b…

Let $\mathcal{Z}[g]$ be the partition function of the Coulomb plasma on a surface of constant curvature, let $N$ be the number of particles, and let $\chi$ be the Euler characteris…