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

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here.

Conventions. By The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particles, The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability is in force; let $c_{1},C_{1}$ be the constants of The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §lower and $a_{0},b_{0}$ those of The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses. $V$ and $V'$ are continuous and Borel by A Confining Potential and Its Derivative are Continuous and Borel §continuous and A Confining Potential and Its Derivative are Continuous and Borel §borel. $\ell$ is the logarithmic kernel, Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel, with diagonal $\Delta$; we write $(s,t)$ for the point $\iota(s,t)$ of $\mathbb{R}^{2}$ given by the concatenation map of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, and $A\times B$ for $\iota(A\times B)$. For $\rho,\rho'\in\mathcal{P}(\mathbb{R})$ we use the following fact (F): for every Borel $F:\mathbb{R}^{2}\to[0,\infty]$

and the same identities hold for $\rho\boxtimes\rho'$-integrable Borel $F:\mathbb{R}^{2}\to\mathbb{R}$, the inner integrals being defined off null sets. This is Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product combined with the Tonelli and Fubini parts of Tonelli and Fubini Theorems and claim 2 of Image Measures, Measures with Densities, and Change of Variables. In particular $(\rho\boxtimes\rho')(A\times B)=\rho(A)\rho'(B)$, the marginals of $\rho\boxtimes\rho'$ are $\rho$ and $\rho'$, and, as $\ell(s,t)=\ell(t,s)$, $\int_{A\times B}\ell\,d(\rho\boxtimes\rho)=\int_{B\times A}\ell\,d(\rho\boxtimes\rho)$ for Borel $A,B$ whenever $\ell$ is $\rho\boxtimes\rho$-integrable. For $x\in\mathbb{R}^{N}$, (F) and Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral (with $q=1$), used twice, give (E): $\int F\,d(\mu^{N}_{x}\boxtimes\mu^{N}_{x})=N^{-2}\sum_{i,j=1}^{N}F(x_{i},x_{j})$ for every Borel $F$ bounded below.

For $x\in W_{N}$ and $i<j$ we have $x_{i}-x_{j}=|x_{i}-x_{j}|>0$ by The Weyl Chamber of Ordered Points in Euclidean Space, so by The Logarithmic Energy of N Ordered Particles on the Weyl Chamber §energy and the symmetry of $\ell$, $H_{b_{N}}(x)=b_{N}\sum_{i<j}\ell(x_{i},x_{j})=\frac{b_{N}}{2}\sum_{i\ne j}\ell(x_{i},x_{j})$. With $b_{N}=\beta/(2(N-1))$ and $S_{x}=N^{-2}\sum_{i\ne j}\ell(x_{i},x_{j})$ this reads

Step 1 (Bounds). By Confining Potentials on the Real Line §superquadratic with $M=1$ there is a positive $K$ with $t^{2}\le V(t)$ for $|t|\ge K$, and by The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses, $V(t)\ge a_{0}|t|-b_{0}\ge-|b_{0}|$ for every $t$. Put $K_{V}=K^{2}+|b_{0}|$. Then

the first for $|t|<K$ because $t^{2}-K_{V}\le-|b_{0}|$ there. Let $e_{*}=-(c_{1}K_{V}+C_{1})$ and $A_{0}=1/c_{1}>0$. For $N\ge2$ and $x\in W_{N}$, The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §lower and (1) give $P_{N}(x)\ge c_{1}\sum_{k}V(x_{k})-C_{1}N\ge-c_{1}K_{V}N-C_{1}N=e_{*}N$, and

This is claim The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds.

Step 2 (Compactness and lower limit). Let $(N_{k})$, $(x^{k})$ and $c$ be as in claim The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §liminf, write $\mu_{k}=\mu^{N_{k}}_{x^{k}}$ and $a_{k}=P_{N_{k}}(x^{k})/N_{k}$, so that $e_{*}\le a_{k}\le c$ by Step 1.

(2a) Compactness. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment and Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral, $\mu_{k}\in\mathcal{P}_{2}(\mathbb{R})$, $V$ is $\mu_{k}$-integrable, and by The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §lower

$V$ is Borel, bounded below by (1), and superquadratic in the sense of Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §compactness (with $m=1$) by Confining Potentials on the Real Line §superquadratic. That clause gives $\nu\in\mathcal{P}_{2}(\mathbb{R})$ and a strictly increasing $(k_{j})$ with $W_{2}(\mu_{k_{j}},\nu)\to0$. Write $\mu_{j}'=\mu_{k_{j}}$, $N_{j}'=N_{k_{j}}$, $a_{j}'=a_{k_{j}}$ and $\ell_{*}=\liminf_{j}a_{j}'$, the limit inferior of a sequence in $[e_{*},c]$; note $N_{j}'\ge j$, so $N_{j}'\to\infty$.

(2b) The squares converge weakly. Let $f:\mathbb{R}^{2}\to\mathbb{R}$ be bounded and Lipschitz with constant $L$. For each $s$, $t\mapsto f(s,t)$ and $t\mapsto f(t,s)$ are bounded and Lipschitz with constant $L$ on $\mathbb{R}$, since $\lVert(s,t)-(s,t')\rVert=|t-t'|$. For every $\pi\in\Pi(\mu_{j}',\nu)$, finite in cost by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite, Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §lipschitz bounds $|\int f(s,t)\mu_{j}'(dt)-\int f(s,t)\nu(dt)|$ by $L\sqrt{I(\pi)}$; taking the infimum over $\pi$ as in The Quadratic Wasserstein Distance on Euclidean Space §distance, the bound $L\,W_{2}(\mu_{j}',\nu)$ follows, and likewise in the other variable. By (F), inserting $\mu_{j}'\boxtimes\nu$,

By claim 1 of Portmanteau Theorem on a Metric Space on $\mathbb{R}^{2}$, $\mu_{j}'\boxtimes\mu_{j}'$ converges weakly to $\nu\boxtimes\nu$.

(2c) Truncated kernels. For a natural number $M\ge1$ let $\ell_{M}(s,t)=\min\{\ell(s,t),M\}$ for $s\ne t$ and $\ell_{M}(s,s)=M$. On the open set $\{|s-t|<e^{-M}\}$, which contains $\Delta$, $\ell_{M}\equiv M$; on the open set $\{s\ne t\}$, $\ell_{M}=\min\{-\log|s-t|,M\}$ is continuous; so $\ell_{M}$ is continuous. By The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §lower-bound and $M>0$, $\ell_{M}(s,t)\ge-|s|-|t|$. Put $\kappa=K_{V}+\beta^{2}/16$ and

continuous, respectively Borel. Since $\frac{\beta}{4}|s|\le\frac{s^{2}}{2}+\frac{\beta^{2}}{32}$, (1) and the two lower bounds for $\ell$ and $\ell_{M}$ give $k_{M}\ge-\kappa$ and $k\ge-\kappa$ off $\Delta$; on $\Delta$, $k_{M}(s,s)\ge V(s)\ge-\kappa$ and $k(s,s)=V(s)\ge-\kappa$. Moreover $k_{M}\le k_{M+1}$, and $\sup_{M}k_{M}=k$ off $\Delta$, $=+\infty$ on $\Delta$.

(2d) Empirical bound. Let $N\ge2$, $x\in W_{N}$ with $P_{N}(x)\le cN$, and $\mu=\mu^{N}_{x}$. By (E), $\ell_{M}\le\ell$ off $\Delta$ and $\sum_{i\ne j}\tfrac12(V(x_{i})+V(x_{j}))=(N-1)\sum_{i}V(x_{i})$,

By The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §lower-bound, $|x_{i}|\le\frac12(1+x_{i}^{2})$ and Step 1, $S_{x}\ge-\frac{2}{N}\sum_{i}|x_{i}|\ge-\bigl(1+\lVert x\rVert^{2}/N\bigr)\ge-s_{0}$ with $s_{0}=1+A_{0}(1+c-e_{*})$. Hence $\frac{N}{N-1}S_{x}\ge S_{x}-\frac{s_{0}}{N-1}$ (trivially if $S_{x}\ge0$), and with (P)

(2e) Passage to the limit. Fix $M$, and a natural number $L\ge1$. The function $\min\{k_{M},L\}+\kappa$ is continuous, with values in $[0,L+\kappa]$, so by (2b) and Weak Convergence of Finite Borel Measures on a Metric Space, its integrals $G_{j}$ against $\mu_{j}'\boxtimes\mu_{j}'$ converge to its integral $G$ against $\nu\boxtimes\nu$; by (2), $G_{j}\le a_{j}'+\varepsilon_{N_{j}',M}+\kappa$. Given $\eta>0$, choose $j$ so large that $G_{j}>G-\eta$, $\varepsilon_{N_{j}',M}<\eta$ and $a_{j}'<\ell_{*}+\eta$ (possible by claim 4 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence); then $G<\ell_{*}+\kappa+3\eta$. Hence $G\le\ell_{*}+\kappa$. Letting $L\to\infty$, Monotone Convergence Theorem gives $\int(k_{M}+\kappa)\,d(\nu\boxtimes\nu)\le\ell_{*}+\kappa$; letting then $M\to\infty$, again by Monotone Convergence Theorem and (2c),

As $\sup_{M}(k_{M}+\kappa)=+\infty$ on $\Delta$, $(\nu\boxtimes\nu)(\Delta)=0$; so $k+\kappa=\sup_{M}(k_{M}+\kappa)$ almost everywhere and $\int(k+\kappa)\,d(\nu\boxtimes\nu)\le\ell_{*}+\kappa$.

(2f) $\nu\in\mathcal{D}$ and the lower limit. For $a\in\mathbb{R}$, $\nu(\{a\})^{2}=(\nu\boxtimes\nu)(\{a\}\times\{a\})\le(\nu\boxtimes\nu)(\Delta)=0$, so $\nu$ is atomless. Let $g(s,t)=\frac12(V(s)+V(t))+K_{V}\ge0$. From $k=\frac{\beta}{4}\ell+g-K_{V}$ and $\ell\ge-|s|-|t|$ we get $0\le g\le(k+\kappa)+\frac{\beta}{4}(|s|+|t|)+K_{V}$, which is integrable because $\nu\in\mathcal{P}_{2}(\mathbb{R})$ and $|s|\le\frac12(1+s^{2})$; by (F), $\int g\,d(\nu\boxtimes\nu)=\int(V+K_{V})\,d\nu$, so $V$ is $\nu$-integrable. Then $\ell=\frac{4}{\beta}(k-g+K_{V})$ is $\nu\boxtimes\nu$-integrable. By The Logarithmic Energy of a Probability Measure on the Real Line §energy, $\nu\in\mathcal{D}_{\log}$, and by The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair, $\nu\in\mathcal{D}$ with, by (F) for $g$,

Given $\varepsilon>0$, claim 3 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence gives $j_{0}$ with $\ell_{*}-\varepsilon<a_{j}'$ for $j\ge j_{0}$, so $\mathcal{E}(\nu)\le P_{N_{k_{j}}}(x^{k_{j}})/N_{k_{j}}+\varepsilon$ for $j\ge j_{0}$. This proves claim The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §liminf.

Step 3 (Recovery). Let $\mu\in\mathcal{D}$ and $\varepsilon>0$; thus $\mu$ is atomless, $\ell$ is $\mu\boxtimes\mu$-integrable and $V$ is $\mu$-integrable.

(3a) Truncation. For a natural number $n\ge1$ let $I_{n}=[-n,n]$ and $p_{n}=\mu(I_{n})$. Since $\mathbf{1}_{I_{n}}\uparrow1$ pointwise, $p_{n}\to1$ by Monotone Convergence Theorem; fix $n_{0}$ with $p_{n}\ge\frac12$ for $n\ge n_{0}$. For such $n$ let $\mu_{n}$ be the measure with density $p_{n}^{-1}\mathbf{1}_{I_{n}}$ with respect to $\mu$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables): a probability measure with $\mu_{n}(\mathbb{R}\setminus I_{n})=0$, $\int h\,d\mu_{n}=p_{n}^{-1}\int h\mathbf{1}_{I_{n}}\,d\mu$ for Borel $h\ge0$ and for $\mu$-integrable $h$, and $\mu_{n}(\{a\})\le2\mu(\{a\})=0$. Hence $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R})$, and by (F) applied twice, $\int F\,d(\mu_{n}\boxtimes\mu_{n})=p_{n}^{-2}\int F\,\mathbf{1}_{I_{n}\times I_{n}}\,d(\mu\boxtimes\mu)$ for Borel $F\ge0$, hence for $F=\ell^{\pm}$. So $\ell$ is $\mu_{n}\boxtimes\mu_{n}$-integrable, $V$ is $\mu_{n}$-integrable, $\mu_{n}\in\mathcal{D}$, and by Dominated Convergence Theorem (dominating functions $|\ell|$ and $|V|$) and $p_{n}\to1$,

so $\mathcal{E}(\mu_{n})\to\mathcal{E}(\mu)$. Moreover $W_{2}(\mu_{n},\mu)\to0$ by Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §convergence applied to $(\mu_{n})_{n\ge n_{0}}$: for bounded continuous $f$, $\int f\,d\mu_{n}=p_{n}^{-1}\int f\mathbf{1}_{I_{n}}\,d\mu\to\int f\,d\mu$ by Dominated Convergence Theorem, which is weak convergence; and given $\eta>0$, Dominated Convergence Theorem applied to $s^{2}\mathbf{1}_{\{|s|>K\}}$ (dominated by $s^{2}$) gives $K$ with $\int_{\{|s|>K\}}s^{2}\,d\mu<\eta/2$, whence $\int_{\{|s|>K\}}s^{2}\,d\mu_{n}\le2\int_{\{|s|>K\}}s^{2}\,d\mu<\eta$ for every $n\ge n_{0}$. Fix $R=n\ge n_{0}$ with

and write $\hat{\mu}=\mu_{R}$, $I=I_{R}$: an atomless member of $\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R})$ with $\hat{\mu}(\mathbb{R}\setminus I)=0$, hence $(\hat{\mu}\boxtimes\hat{\mu})(\mathbb{R}^{2}\setminus I\times I)=0$ by (F).

(3b) Block means. Fix $N\ge2$ and let $B_{1},\dots,B_{N}$ be the quantile blocks of $\hat{\mu}$ at level $N$ from Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function; by Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §blocks they partition $\mathbb{R}$ into Borel sets of mass $\frac1N$, and $t<s$ whenever $s\in B_{i}$, $t\in B_{j}$, $i<j$. Put $y_{i}=N\int_{B_{i}}s\,\hat{\mu}(ds)$ (the block average $m_{i}$ of the lemma; $\mathrm{id}$ is $\hat{\mu}$-integrable as $|s|\le R$ a.e.). Then $|y_{i}|\le N\int_{B_{i}}|s|\,d\hat{\mu}\le R$. For $i<j$, (F) gives

the integrand being positive on $B_{i}\times B_{j}$, a set of measure $N^{-2}>0$. So $y$ is ordered, indeed strictly decreasing.

(3c) Distance. Let $J_{i}=B_{i}\cap I$, nonempty as $\hat{\mu}(J_{i})=\frac1N$, with $a_{i}=\inf J_{i}$, $b_{i}=\sup J_{i}$ and $w_{i}=b_{i}-a_{i}\ge0$. As $y_{i}=N\int_{J_{i}}s\,d\hat{\mu}$ is an average of points of $[a_{i},b_{i}]$, $|y_{i}-s|\le w_{i}$ for $s\in J_{i}$, while $\hat{\mu}(B_{i}\setminus J_{i})=0$. By the ordering of blocks, $b_{i+1}\le a_{i}$, so $\sum_{i}w_{i}\le b_{1}-a_{N}\le2R$. By Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal,

(3d) Energy of the block means. (i) For $u,m>0$, $-\log u\ge-\log m-\frac{u-m}{m}$: with $z=u/m$ and $\log u-\log m=\log z$ (The Natural Logarithm), for $z\ne1$ Mean Value Theorem on a Closed Real Interval on the interval with endpoints $1,z$ gives $\log z=(z-1)/\xi$ with $\xi$ strictly between $1$ and $z$, and $(z-1)/\xi<z-1$ in both cases $z>1$ and $z<1$. Let $i<j$ and $m=y_{i}-y_{j}>0$. On $B_{i}\times B_{j}$, $\ell(s,t)=-\log(s-t)\ge-\log m-\frac{(s-t)-m}{m}$; integrating against $\hat{\mu}\boxtimes\hat{\mu}$ ($\ell$ being integrable) and using (4),

(ii) Let $\lambda_{R}=\max\{\log(2R),0\}$. On $I\times I$, $\ell\ge-\lambda_{R}$ (off $\Delta$, $|s-t|\le2R$ and $\log$ is increasing; on $\Delta$, $\ell=0$), so $\int_{B_{i}\times B_{i}}\ell\,d(\hat{\mu}\boxtimes\hat{\mu})\ge-\lambda_{R}N^{-2}$. (iii) The sets $B_{i}\times B_{j}$ partition $\mathbb{R}^{2}$, so by the symmetry recorded in (F),

(iv) Tangent inequality: for $u,u'\in\mathbb{R}$, $V(u')\ge V(u)+V'(u)(u'-u)$. Indeed, by convexity, for $0<\theta\le1$, $V(u+\theta(u'-u))-V(u)\le\theta(V(u')-V(u))$; for $u'\ne u$ divide by $\theta$ and let $\theta\to0$, the left side tending to $V'(u)(u'-u)$ by Derivative at an Interior Point. Applied with $u=y_{i}$ and integrated against $N\hat{\mu}$ over $B_{i}$ ($V$ being $\hat{\mu}$-integrable), it gives $V(y_{i})\le N\int_{B_{i}}V\,d\hat{\mu}$, so $\sum_{i}V(y_{i})\le N\int V\,d\hat{\mu}$.

(3e) Spreading and choice of constants. By Extreme Value Theorem on a Closed Real Interval applied to the continuous $|V'|$ on $[-R-1,R+1]$ there is $L_{R}\ge0$ bounding $|V'|$ there; by the tangent inequality at $u$, $V(u)-V(u')\le L_{R}|u-u'|$ for $u,u'\in[-R-1,R+1]$. Let

Then $y'_{i}-y'_{i+1}=y_{i}-y_{i+1}+\frac{\delta}{N}\ge\frac{c}{N}>0$, so $y'\in W_{N}$; $|y'_{i}|\le R+\delta\le R'$; $|y'_{i}-y_{i}|\le\delta$, so $W_{2}(\mu^{N}_{y'},\mu^{N}_{y})\le\delta\le\varepsilon/3$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §lipschitz; $V(y'_{i})\le V(y_{i})+L_{R}\delta$ with $L_{R}\delta<\varepsilon/3$; and for $i<j$, $y'_{i}-y'_{j}\ge y_{i}-y_{j}>0$, so $-\log(y'_{i}-y'_{j})\le-\log(y_{i}-y_{j})$. By The Logarithmic Energy of N Ordered Particles on the Weyl Chamber §energy, (6), (3d)(iv) and $b_{N}N^{2}/(2N)=\frac{\beta}{4}\frac{N}{N-1}$, with $e_{N}=\mathcal{E}_{\log}(\hat{\mu})+\lambda_{R}/N$ and $\frac{N}{N-1}e_{N}\le e_{N}+\frac{|e_{N}|}{N-1}$,

Choose $N_{1}\ge2$ with $2R/\sqrt{N}\le\varepsilon/3$ and $\frac{\beta}{4}(|\mathcal{E}_{\log}(\hat{\mu})|+2\lambda_{R})/(N-1)\le\varepsilon/3$ for all $N\ge N_{1}$. For such $N$ and $y'$ as above, (3) gives $P_{N}(y')<N(\mathcal{E}(\mu)+\varepsilon)$, and (5), (3) and the triangle inequality of The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle give

With $y'$ in place of $y$ and the positive constants $c$ and $R'$ in place of $c$ and $R$, this is claim The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery.