MathOverflow

Let
$$V(z)=\sum_{j=1}^n G_{\mathbb D}(z,a_j),\qquadG_{\mathbb D}(z,a)=\log\left|\frac{1-\overline a z}{z-a}\right|.$$
I am trying to understand whether such a finite Green potential can remain large after passing through many conformal bottlenecks.

Let $L = \{x_1,\ldots,x_n\} \subset [0,1] \cap \mathbb{Q}$ be a finite set of rationally encoded points on the unit interval, and let $s > 0$. For a finite subset $S \subseteq L$, define its Riesz $s$-energy by
$$E_s(S) = \sum_{\{i,j\} \subseteq S} \frac{1}{|x_i-x_j|^s}.$$...

Let $\kappa_\alpha(x_1,\dots,x_n)$ denote the key polynomial, or Demazure character, indexed by a weak composition
$$\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb Z_{\ge 0}^n.$$
For a polynomial $f$, write $\operatorname{Supp}(f)$ for the set of exponent vectors of monomials appearing in $f$ with nonzero coefficient....

Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a positive-definite matrix such that $\operatorname{tr} \Sigma...