Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations, with $N_{0}$, $e_{*}$, $\overline{w}_{N,\tau}$, $\underline{w}_{N,\tau}$, $\bar{u}_{\tau}$ and $\underline{u}_{\tau}$ as in Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy. For $N\ge2$, the set $\mathcal{D}_{N}$, the relative free energy $\mathcal{E}_{N}$, the set $\mathcal{D}^{\Sigma}_{N}$ and the relative score $\Sigma_{N}$ are those of The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set read at the configuration level with $D=W_{N}$, $U=P_{N}$ and $a=a_{N}$. Fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$ (Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below, the pair being Wasserstein-coercive by The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §coercive). Optimal maps from $\hat{\mu}$ to $\nu\in\mathcal{P}_{2}(\mathbb{R})$ are those of that definition; the class of such a map $T$ lies in $L^{2}(\hat{\mu};\mathbb{R})$ and $\lVert T-\mathrm{id}\rVert_{\hat{\mu}}=W_{2}(\nu,\hat{\mu})$ (Step 1 of the proof).

Data. Let $\tau,\delta\in\mathbb{R}$ with $0<\tau$ and $0<\delta<1$, let $\hat{\mu}\in\mathcal{D}$ (an atomless measure, by The Logarithmic Energy of a Probability Measure on the Real Line §energy), let $q\in L^{2}(\hat{\mu};\mathbb{R})$, let $\eta',r_{0},K\in\mathbb{R}$ with $0\le\eta'$, $0<r_{0}$, $1\le K$ and

For $N\ge N_{0}$ let $\chi^{+}_{N},\chi^{-}_{N}:\mathbb{R}^{N}\to\mathbb{R}$ be the block test functions of $\hat{\mu}$ at level $N$ with data $(q,K)$ and $(q,-K)$ respectively.

1. (Subsolution side)¶ Assume that for every $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<r_{0}$ and every optimal map $T$ from $\hat{\mu}$ to $\nu$

and let $(N_{k})_{k}$ be a sequence of realising levels for $\bar{u}_{\tau}$ at $\hat{\mu}$. For each $k$ put $N=N_{k}$, let $f_{k}=\delta^{-1}\bigl(\overline{w}_{N,\tau}-N\chi^{+}_{N}\bigr)$ on $W_{N}$, and let $\pi_{k}=\pi_{f_{k}}$ be the Gibbs measure of Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §gibbs with $D=W_{N}$, $U=P_{N}$, $a=a_{N}$ and $f=f_{k}$, whose hypotheses hold (Step 1 of the proof); thus, by Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §variational, $\pi_{k}$ maximises $P\mapsto\int(\overline{w}_{N,\tau}-N\chi^{+}_{N})\,dP-\delta\,\mathcal{E}_{N}(P)$ over the $P\in\mathcal{D}_{N}$ for which the integrand is $P$-integrable. Then: (a)¶ (a consequence of Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §score) $\pi_{k}\in\mathcal{D}^{\Sigma}_{N}$ and, for every gradient map $\nabla\overline{w}_{N,\tau}$ of $\overline{w}_{N,\tau}$, $\nabla\overline{w}_{N,\tau}=N\,\nabla\chi^{+}_{N}+\delta\,\Sigma_{N}(\pi_{k})$ in $L^{2}(\pi_{k};\mathbb{R}^{N})$; (b)¶ there is $k_{0}$ such that for every $k\ge k_{0}$

(c)¶ for every positive $\varepsilon\in\mathbb{R}$ there is $k_{0}$ with $K\int W_{2}(\mu^{N}_{x},\hat{\mu})^{2}\,\pi_{k}(dx)\le\frac{2\eta'^{2}}{K}+\varepsilon$ for every $k\ge k_{0}$.

2. (Supersolution side)¶ Assume that for every $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<r_{0}$ and every optimal map $T$ from $\hat{\mu}$ to $\nu$

let $(N_{k})_{k}$ be a sequence of realising levels for $\underline{u}_{\tau}$ at $\hat{\mu}$, and let $\pi_{k}$ be as in clause 1 with $f_{k}=\delta^{-1}\bigl(N\chi^{-}_{N}-\underline{w}_{N,\tau}\bigr)$; thus $\pi_{k}$ minimises $P\mapsto\int(\underline{w}_{N,\tau}-N\chi^{-}_{N})\,dP+\delta\,\mathcal{E}_{N}(P)$ over the same class. Then (a) holds with $\nabla\underline{w}_{N,\tau}=N\,\nabla\chi^{-}_{N}-\delta\,\Sigma_{N}(\pi_{k})$ for every gradient map of $\underline{w}_{N,\tau}$, and (b) and (c) hold as stated.