The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations. For a natural number $N\ge2$, the set $\mathcal{D}^{\Sigma}_{N}$ and the relative score $\Sigma_{N}(P)=\nabla P_{N}+a_{N}\xi_{P}\in L^{2}(P;\mathbb{R}^{N})$ of $P\in\mathcal{D}^{\Sigma}_{N}$ are the set $\mathcal{D}^{\Sigma}_{U,a}$ and the relative score 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 $a=a_{N}$; here $\lVert\cdot\rVert_{P}$ and $\langle\cdot,\cdot\rangle_{P}$ are the norm and inner product of $L^{2}(P;\mathbb{R}^{N})$. For $\psi\in C_{c}^{\infty}(\mathbb{R})$ with derivative $\psi'$, let $\psi'^{\oplus}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ be $\psi'^{\oplus}(x)=(\psi'(x_{1}),\dots,\psi'(x_{N}))$.

Data. Let $(N_{k})_{k\in\mathbb{N}}$ be a strictly increasing sequence of natural numbers with $N_{1}\ge2$, let $P_{k}\in\mathcal{D}^{\Sigma}_{N_{k}}$ for every $k$, let $\hat{\mu}\in\mathcal{D}$, and let $A\in\mathbb{R}$ satisfy

the integrand is a Borel function of $x$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §distance.

1. (The limit has a score)¶ $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}^{2}\le A$.

2. (Convergence of pairings)¶ Let $q\in L^{2}(\hat{\mu};\mathbb{R})$ and $G_{k}\in L^{2}(P_{k};\mathbb{R}^{N_{k}})$ for $k\in\mathbb{N}$ be such that for every positive $\varepsilon\in\mathbb{R}$ there are $\psi\in C_{c}^{\infty}(\mathbb{R})$ and $k_{0}\in\mathbb{N}$ with

Then the real sequences $\bigl(\frac{1}{N_{k}}\langle\Sigma_{N_{k}}(P_{k}),G_{k}\rangle_{P_{k}}\bigr)_{k}$ and $\bigl(\frac{1}{N_{k}}\lVert G_{k}\rVert_{P_{k}}^{2}\bigr)_{k}$ converge to $\langle\Sigma(\hat{\mu}),q\rangle_{\hat{\mu}}$ and to $\lVert q\rVert_{\hat{\mu}}^{2}$ respectively.

3. (Lower limit of the Fisher term)¶ For every positive $\varepsilon\in\mathbb{R}$ there is $k_{0}\in\mathbb{N}$ with $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}^{2}\le\frac{1}{N_{k}}\lVert\Sigma_{N_{k}}(P_{k})\rVert_{P_{k}}^{2}+\varepsilon$ for every $k\ge k_{0}$.