The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations; in particular $P_{N}$ is the $N$-particle potential on the Weyl chamber $W_{N}$, $\mu^{N}_{x}$ the empirical measure of $x\in\mathbb{R}^{N}$, and $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ the confined logarithmic-energy pair (clauses The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particles and The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation).

1. (Bounds)¶ There are $e_{*},A_{0}\in\mathbb{R}$ with $0\le A_{0}$ such that for every natural number $N\ge2$ and every $x\in W_{N}$

2. (Compactness and lower limit)¶ Let $(N_{k})_{k\in\mathbb{N}}$ be a strictly increasing sequence of natural numbers with $N_{1}\ge2$, let $x^{k}\in W_{N_{k}}$, and let $c\in\mathbb{R}$ satisfy $P_{N_{k}}(x^{k})\le cN_{k}$ for every $k$. Then there are a strictly increasing sequence $(k_{j})_{j\in\mathbb{N}}$ of natural numbers and $\nu\in\mathcal{D}$ such that $\bigl(W_{2}(\mu^{N_{k_{j}}}_{x^{k_{j}}},\nu)\bigr)_{j}$ converges to $0$ and, for every positive $\varepsilon\in\mathbb{R}$, there is $j_{0}$ with

3. (Recovery with separated particles)¶ Let $\mu\in\mathcal{D}$ and let $\varepsilon\in\mathbb{R}$ be positive. Then there are positive $c,R\in\mathbb{R}$ and a natural number $N_{1}\ge2$ such that for every natural number $N\ge N_{1}$ there is $y\in W_{N}$ with