Upper and Lower Half-Relaxed Limits along Empirical Measures of Functions on the Weyl Chambers

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations, with the Weyl chambers $W_{N}$, the potentials $P_{N}$ and the empirical measures $\mu^{N}_{x}$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particles and the domain $\mathcal{D}$ and energy $\mathcal{E}$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation. Let $N_{0}\ge2$ be a natural number, for every natural number $N\ge N_{0}$ let $f_{N}:W_{N}\to\mathbb{R}$, and let $c\in\mathbb{R}$ be nonnegative. Fix $e_{*}$ as in The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds. For $\mu\in\mathcal{D}$, positive $\rho\in\mathbb{R}$ and a natural number $N_{1}\ge N_{0}$ let

This set is nonempty: by The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery, applied with $\varepsilon$ the least of $\rho$ and $1$, there are a natural number $N'$ and, for $N$ the greatest of $N_{1}$ and $N'$, a point $y\in W_{N}$ with $W_{2}(\mu^{N}_{y},\mu)<\rho$ and $e_{*}N\le P_{N}(y)\le N(\mathcal{E}(\mu)+1)$, the first inequality by The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds; hence $|P_{N}(y)|\le N\,b_{\mu}$ with $b_{\mu}=|e_{*}|+|\mathcal{E}(\mu)|+1$.

1. (Upper half-relaxed limit)¶ Suppose that $-c\bigl(N+|P_{N}(x)|\bigr)\le f_{N}(x)\le cN$ for every $N\ge N_{0}$ and $x\in W_{N}$. Then each $S(\mu,\rho,N_{1})$ is bounded above by $c$ and has a least upper bound $s^{+}(\mu,\rho,N_{1})$, which is at least $\frac{f_{N}(y)}{N}\ge-c(1+b_{\mu})$ for the $N$ and $y$ above. The upper half-relaxed limit of $(f_{N})_{N\ge N_{0}}$ is the function $\bar{f}:\mathcal{D}\to\mathbb{R}$ whose value at $\mu$ is the greatest lower bound of the set of the numbers $s^{+}(\mu,\rho,N_{1})$ over all positive $\rho$ and all natural $N_{1}\ge N_{0}$, a set bounded below by $-c(1+b_{\mu})$.

2. (Lower half-relaxed limit)¶ Suppose that $-cN\le f_{N}(x)\le c\bigl(N+|P_{N}(x)|\bigr)$ for every $N\ge N_{0}$ and $x\in W_{N}$. Then each $S(\mu,\rho,N_{1})$ is bounded below by $-c$ and has a greatest lower bound $s^{-}(\mu,\rho,N_{1})\le c(1+b_{\mu})$. The lower half-relaxed limit of $(f_{N})_{N\ge N_{0}}$ is the function $\underline{f}:\mathcal{D}\to\mathbb{R}$ whose value at $\mu$ is the least upper bound of the numbers $s^{-}(\mu,\rho,N_{1})$ over all positive $\rho$ and all natural $N_{1}\ge N_{0}$, a set bounded above by $c(1+b_{\mu})$.

Least upper and greatest lower bounds exist by the Dedekind completeness of the real numbers.