Proof of The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. Elementary order and field arithmetic of real numbers (rearranging finite sums, multiplying an inequality by a nonnegative number, $2|u||v|\le u^{2}+v^{2}$, $(u+v)^{2}\le2u^{2}+2v^{2}$ and the like) is used without further comment, by Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field; finite sums are sums over finite index sets, and are reindexed along bijections and split over disjoint unions of index sets without comment.

Step 0 (Constants and elementary facts). Put $\gamma=\beta/\sigma^{2}$. Since $0<\sigma^{2}<\beta$, we have $1<\gamma$, and $\varepsilon=1-\sigma^{2}/\beta$ satisfies $0<\varepsilon<1$. For a natural number $N\ge2$ we have $N\ge2$ as real numbers by claims 2 and 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so $N-1\ge1$, and directly from the definitions

By claims 1, 2 and 4 of Basic Properties of the Exponential Function, $\exp(u+v)=\exp(u)\exp(v)$, $\exp(0)=1$, $\exp(u)>0$, $\exp(-u)=1/\exp(u)$ and $\exp$ is strictly increasing; by induction $\exp(ku)=\exp(u)^{k}$ for every natural number $k$. By The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §exp, $1+u\le\exp(u)$ for every real $u$, so also $u\le\exp(u)$. By The Natural Logarithm, $\log$ is the inverse of $\exp$ and $\log(st)=\log s+\log t$; by claim 2 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, $\log$ is strictly increasing, $\log1=0$, and $\log t\ge0$ exactly when $t\ge1$; by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log, $1-t^{-1}\le\log t\le t-1$ for positive $t$. Consequently, for positive $t$, $1/t=\exp(-\log t)$, $1/t^{2}=\exp(-2\log t)$, and

because $\log t\le t-1\le t$ and $-\log t\le t^{-1}-1\le t^{-1}$. For $t\in\mathbb{R}$ put $\rho(t)=1+|t|$ and $\ell(t)=\log\rho(t)$; then $\rho(t)=\exp(\ell(t))$ and $0\le\ell(t)\le\rho(t)-1=|t|$. Finally, for positive $\delta$ and real $t$,

since $\delta t^{2}-|t|+\frac{1}{4\delta}=\delta\bigl(|t|-\frac{1}{2\delta}\bigr)^{2}\ge0$, using $|t|^{2}=t^{2}$.

Step 1 (Clause 1). (a) Regularity. By Confining Potentials on the Real Line §confining, $V$ is differentiable at every point of $\mathbb{R}$ with derivative $V'$, and $V'$ is differentiable at every point with a continuous derivative $V''$. By claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, for every $t\in\mathbb{R}$ the partial derivative $\partial_{1}V(t)$ exists and equals $V'(t)$, and $\partial_{1}V'(t)$ exists and equals $V''(t)$. The functions $V$ and $V'$ are continuous by A Confining Potential and Its Derivative are Continuous and Borel §continuous, and $V''$ is continuous by Confining Potentials on the Real Line §confining; continuity for the absolute-value metric at a point is the continuity of Continuity at a Point for Maps Between Euclidean Spaces there, by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §equivalent. The set $\mathbb{R}=\mathbb{R}^{1}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Hence, by clause 1 of C^k Maps on a Euclidean Open Set, $V'$ is of class $C^{1}$ on $\mathbb{R}$ with $\partial_{1}V'=V''$, and $V$ is of class $C^{1}$ with $\partial_{1}V=V'$; by clause 2 there (with $k=1$), $V$ is of class $C^{2}$ on $\mathbb{R}$, and $\partial_{1}\partial_{1}V=V''$ in the notation of clause 4 there.

(b) $V$ as a confining potential on $\mathbb{R}^{1}$. We check that $V$ is a confining potential on $\mathbb{R}^{d}$ with $d=1$. It is of class $C^{2}$ by (a), and convex on $\mathbb{R}=\mathbb{R}^{1}$ in the sense of Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ with $n=1$ by Confining Potentials on the Real Line §confining. For $x\in\mathbb{R}^{1}$, $\lVert x\rVert^{2}=x^{2}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $\lVert x\rVert=|x|$, both being nonnegative with the same square (Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); the gradient $DV(x)$ is the point $(\partial_{1}V(x))=(V'(x))$ of $\mathbb{R}^{1}$, so $\lVert DV(x)\rVert=|V'(x)|$; the Laplacian is $\Delta V(x)=\partial_{1}\partial_{1}V(x)=V''(x)$; and the dot product of $x,y\in\mathbb{R}^{1}$ is $xy$. With these readings, conditions (a), (b) and (c) of Confining Potentials on Euclidean Space §confining are Confining Potentials on the Real Line §superquadratic, Confining Potentials on the Real Line §slope and Confining Potentials on the Real Line §curvature. Therefore Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability applies with $d=1$ and gives: by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §minorant, for every positive $M$ there is $C_{M}\in\mathbb{R}$ with

by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §tangent,

and by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §hessian,

Fix the constant $C_{1}'$ of (1.1) for $M=1$ and put $A_{0}=|C_{1}'|$. Then for every $t\in\mathbb{R}$

the last because $|V(t)|=V(t)$ if $V(t)\ge0$, while otherwise $|V(t)|=-V(t)\le V(t)+2A_{0}$ as $-A_{0}\le V(t)$.

(c) Linear lower bound. By (0.3) with $\delta=1$ and by (1.4), $|t|\le t^{2}+\frac14\le V(t)+A_{0}+\frac14$; so $a_{0}=1$ and $b_{0}=A_{0}+\frac14$ satisfy $0<a_{0}$ and $a_{0}|t|-b_{0}\le V(t)$ for every $t$.

(d) Regular growth. Let $t\in\mathbb{R}$. By (1.2) with $s=t$ and $0$ in place of $t$, $V(t)-tV'(t)\le V(0)$, so by (1.4)

whence $V(t)\le V'(t)^{2}+2V(0)+A_{0}$ and, by (1.4), $|V(t)|\le V'(t)^{2}+K_{V}$ with $K_{V}=2|V(0)|+3A_{0}$. Now let $\eta$ be positive and let $C_{\eta}'$ be the constant of Confining Potentials on the Real Line §curvature for $\eta$ in place of $\varepsilon$ there. Then for every $t$

which is regular growth with $C_{\eta}=\eta K_{V}+C_{\eta}'$.

(e) Consequences. The hypotheses of Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold on $V_{1}$ are that $V_{1}$ is of class $C^{2}$ on $\mathbb{R}$, that $V_{1}''\ge0$, that $a_{0}|t|-b_{0}\le V_{1}(t)$ for some $a_{0}>0$ and $b_{0}$, and that $V_{1}$ has regular growth, where $V_{1}'=\partial_{1}V_{1}$ and $V_{1}''=\partial_{1}\partial_{1}V_{1}$ as in The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality. For $V_{1}=V$ these derivatives are $V'$ and $V''$ by (a), and the hypotheses hold by (a), (1.3), (c) and (d). Since $V$ is of class $C^{2}$, The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality may be used with strength $b_{N}$ in place of $\beta$ and $V_{1}=V$, and its function $P=H_{b_{N}}+\sum_{k}V(x_{k})$ is $P_{N}$. This proves clause 1.

Step 2 (Clause 2). Let $N\ge2$. By Step 1(e) and (c), the hypothesis of The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §penalty holds for strength $b_{N}$ and $V_{1}=V$, so $P_{N}$ is a penalty on $W_{N}$; by (1.3) the hypothesis of The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §monotone holds, which gives $0\le(DP_{N}(x)-DP_{N}(y))\cdot(x-y)$ for all $x,y\in W_{N}$. Finally $\kappa_{N}=\sigma^{2}/(N-1)$ is positive and $2b_{N}=\beta/(N-1)$; as $\sigma^{2}<\beta$ and $(N-1)^{-1}>0$, $\kappa_{N}<2b_{N}$. This proves clause 2.

Step 3 (Notation on the chamber). In Steps 3 to 6 fix a natural number $N\ge2$ and write $b=b_{N}$, $a=a_{N}$, $H=H_{b}$ and $P=P_{N}$; let $\Pi=\{(i,j)\in[N]\times[N]:i<j\}$ be the index set of The Logarithmic Energy of N Ordered Particles on the Weyl Chamber, and let $a_{kj}(x)$ and $S(x)$ be as in The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity (they do not depend on the strength). Constants introduced in Steps 4 and 5 will not depend on $N$. For $x\in W_{N}$ and $(i,j)\in\Pi$ put $s_{ij}=x_{i}-x_{j}$, which is positive by The Logarithmic Energy of N Ordered Particles on the Weyl Chamber; for $k\in[N]$ put $\rho_{k}=\rho(x_{k})$ and $\ell_{k}=\ell(x_{k})$, and put $L(x)=\sum_{k=1}^{N}\ell_{k}\ge0$.

For $(i,j)\in\Pi$, $s_{ij}\le|x_{i}|+|x_{j}|\le\rho_{i}\rho_{j}$, hence, as $\log$ is increasing and $\log(\rho_{i}\rho_{j})=\ell_{i}+\ell_{j}$,

For real numbers $u_{1},\dots,u_{N}$,

indeed $(i,j)\mapsto(j,i)$ maps $\Pi$ bijectively onto $\Pi'=\{(i,j)\in[N]\times[N]:j<i\}$, the sets $\Pi$ and $\Pi'$ partition the set $\Pi^{\ne}$ of pairs $(i,j)\in[N]\times[N]$ with $i\ne j$, so the left side is $\sum_{(i,j)\in\Pi^{\ne}}u_{i}$, and each $i\in[N]$ occurs in exactly $N-1$ pairs of $\Pi^{\ne}$. By the same partition, $\sum_{(k,j)\in\Pi^{\ne}}c_{kj}=2\sum_{(i,j)\in\Pi}c_{ij}$ whenever $c_{kj}=c_{jk}$; $\Pi^{\ne}$ has $N(N-1)$ elements and $\Pi$ at most $N^{2}$. Since $a_{kk}(x)=0$ and $a_{kj}(x)^{2}=(x_{k}-x_{j})^{-2}=a_{jk}(x)^{2}$ for $k\ne j$,

By The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §regularity and The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §derivatives (with strength $b$), for $x\in W_{N}$ and $k\in[N]$

Let $z=(V'(x_{1}),\dots,V'(x_{N}))\in\mathbb{R}^{N}$. By (3.4) and Gradient of a Real-Valued Function on a Euclidean Open Set, $DP(x)=z+DH(x)$, so by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, $\lVert DP(x)\rVert^{2}=\lVert z\rVert^{2}+2\,DH(x)\cdot z+\lVert DH(x)\rVert^{2}$. By The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §symmetrisation (strength $b$, with this $z$), $2\,DH(x)\cdot z=-b\,T(x)$ with

and by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §calogero (strength $b$), $\lVert DH(x)\rVert^{2}=b^{2}S(x)$. Hence

Also, applying $(u+v)^{2}\le2u^{2}+2v^{2}$ coordinatewise to $DP(x)=z+DH(x)$ and using the Calogero identity again,

Step 4 (Clause 4). Let $x\in W_{N}$. By The Logarithmic Energy of N Ordered Particles on the Weyl Chamber §energy, (3.1), (3.2) and (0.1), since $-b<0$,

For each $k$, by Step 0, (0.3) with $\delta=1/\beta$, and (1.4), $\ell_{k}\le|x_{k}|\le\frac{1}{\beta}x_{k}^{2}+\frac{\beta}{4}\le\frac{1}{\beta}\bigl(V(x_{k})+A_{0}\bigr)+\frac{\beta}{4}$. Therefore $H(x)\ge-\frac12\sum_{k}V(x_{k})-N\bigl(\frac{A_{0}}{2}+\frac{\beta^{2}}{8}\bigr)$, and

The constants $c_{1}=\frac12>0$ and $C_{1}\ge0$ depend only on $\beta$ and $V$, not on $N$ or $x$. This proves clause 4. Combining (4.1) with (1.4),

Step 5 (Clause 3). (a) A difference-quotient bound. Let $\eta$ be positive, with $C_{\eta}'$ as in Step 1(d). We claim that for all real $s\ne t$

Let $u<v$ be the two numbers $s,t$ in increasing order. The restriction of $V'$ to the closed interval $[u,v]$ is continuous on $[u,v]$, $V'$ being continuous. At each $c\in(u,v)$ the number $V''(c)$ satisfies the defining condition of Derivative at an Interior Point for the restriction on the interval $[u,v]$, because that condition for $V'$ on the interval $\mathbb{R}$ only becomes weaker when the increments $h$ are restricted by $c+h\in[u,v]$; and two numbers satisfying that condition at the interior point $c$ coincide, since for every positive $\varepsilon'$ both lie within $\varepsilon'$ of one difference quotient with an admissible $h$, which exists as $c$ is interior. So the restriction is differentiable at $c$ with derivative $V''(c)$. By Mean Value Theorem on a Closed Real Interval there is $c\in(u,v)$ with $V''(c)=(V'(v)-V'(u))/(v-u)$, which is the quotient in (5.1). It is nonnegative by (1.3), and by Confining Potentials on the Real Line §curvature, $V''(c)\le\eta|V(c)|+C_{\eta}'$. With $\tau=(c-t)/(s-t)$ we have $0<\tau<1$ and $c=\tau s+(1-\tau)t$, so by the convexity of $V$ (Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$, as in Step 1(b)) $V(c)\le\tau V(s)+(1-\tau)V(t)\le|V(s)|+|V(t)|$, and by (1.4) $|V(c)|\le V(c)+2A_{0}\le|V(s)|+|V(t)|+2A_{0}$. This gives (5.1).

(b) The dissipation inequality. Fix $\eta=\lambda/(3\sigma^{2})$ and put $C^{+}=|C_{\eta}'|$; both depend only on $\lambda$, $\sigma$ and $V$. Let $x\in W_{N}$. By (3.4), (3.5), $\kappa_{N}/2=a$ and (0.1),

so that the terms in $S(x)$ cancel and

In $T(x)$ the terms with $k=j$ vanish, as $a_{kk}(x)=0$, and for $k\ne j$ the term $a_{kj}(x)(V'(x_{k})-V'(x_{j}))$ is the quotient of (5.1) with $s=x_{k}$, $t=x_{j}$. Summing (5.1) over $\Pi^{\ne}$ and using the symmetric-sum identity of Step 3 together with (3.2),

by (0.1). By (1.3) and Confining Potentials on the Real Line §curvature, $0\le V''(x_{k})\le\eta|V(x_{k})|+C^{+}$, so, as $a\le\sigma^{2}/2$, $a\sum_{k}V''(x_{k})\le\frac{\sigma^{2}}{2}\bigl(\eta\sum_{k}|V(x_{k})|+NC^{+}\bigr)$. Adding, and using $\frac{3\sigma^{2}\eta}{2}=\frac{\lambda}{2}$, $\sigma^{2}\eta=\frac{\lambda}{3}$ and (4.2),

with $C_{0}=\lambda C_{1}+\frac{4}{3}\lambda A_{0}+\sigma^{2}C^{+}\ge0$, which depends only on $\beta$, $\sigma$, $\lambda$ and $V$. Together with (5.2) this is clause 3.

Step 6 (Clause 5). Keep $N$, $b$, $a$, $H$, $P$ as in Step 3, write $g=g_{N}$, and let $\psi(y)=\exp(-\lVert y\rVert^{2})$ for $y$ in $\mathbb{R}^{N}$ or in $\mathbb{R}^{N-1}$; this is the function $\psi_{\eta}$ of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $\eta=\frac12$, so by claim 1 there (with $m=N$ and with $m=N-1$) it is measurable and $\int\psi\,d\mathcal{L}^{m}<\infty$, the measure written $\lambda_{m}$ there being $\mathcal{L}^{m}$.

(a) Sign and measurability. For $x\in W_{N}$ the bracket in $g(x)$ is at least $1$ and the exponential is positive, so $g(x)>0$; off $W_{N}$, $g=0$. For a map $\varphi$ from a set $E\subseteq\mathbb{R}^{N}$ to $\mathbb{R}$ and $x\in E$, continuity at $x$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces is the same as continuity at $x$ relative to $E$ from $(\mathbb{R}^{N},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$, because for $y\in E$ and positive $\delta,\varepsilon'$ one has $\sum_{i}(y_{i}-x_{i})^{2}<\delta^{2}$ exactly when $d_{E}(y,x)<\delta$, and $(\varphi(y)-\varphi(x))^{2}<\varepsilon'^{2}$ exactly when $|\varphi(y)-\varphi(x)|<\varepsilon'$ (claims 1 and 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). Since $P$ is of class $C^{2}$ on $W_{N}$ (The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §regularity), $P$ and every $\partial_{k}P$ are of class $C^{1}$ and hence continuous at every point of $W_{N}$ (clauses 1 and 2 of C^k Maps on a Euclidean Open Set). The coordinate maps $y\mapsto y_{k}$ are continuous, as $|y_{k}-x_{k}|\le d_{E}(y,x)$ by claims 2 and 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$; the absolute value is continuous on $\mathbb{R}$, as $||u|-|v||\le|u-v|$; and $\exp$ is continuous at every point of $\mathbb{R}$, being differentiable there (claim 3 of Basic Properties of the Exponential Function; Differentiability at an Interior Point Implies Continuity There; Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §equivalent). Since $\lVert DP(y)\rVert^{2}=\sum_{k}\partial_{k}P(y)^{2}$ and $\lVert y\rVert^{2}=\sum_{k}y_{k}^{2}$, it follows from Continuity of Sums and Products of Real-Valued Functions on a Metric Space and Composition of Continuous Euclidean Maps (for $|P|$ and $\exp(-P/a)$) that the restriction of $g$ to $W_{N}$ is continuous at every point of $W_{N}$ relative to $W_{N}$. Now let $c\in\mathbb{R}$. If $c<0$ then $\{g>c\}=\mathbb{R}^{N}$. If $c\ge0$ then $U=\{y:g(y)>c\}\subseteq W_{N}$, and for $x\in U$ there are a positive $\delta$ with $|g(y)-g(x)|<g(x)-c$ for all $y\in W_{N}$ with $d_{E}(y,x)<\delta$, and, $W_{N}$ being open (The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open), a positive $r$ with $B(x,r)\subseteq W_{N}$; then $B(x,\min(\delta,r))\subseteq U$. So $U$ is open, hence (Second-Order Equations on Euclidean Open Sets §space, claims 4 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets) $U\in\mathcal{B}(\mathbb{R}^{N})$. Thus $g$ is measurable in the sense of Lebesgue Integral of a Nonnegative Measurable Function, and, being real-valued, also Borel measurable, as remarked there.

(b) A one-dimensional majorant. Put $\theta=\max(2-\gamma,0)$; as $1<\gamma$, $0\le\theta<1$. Since $2>1$, $\log2>0$. Put $q=\exp(\theta\log2)\ge1$ and $r_{q}=q/2=\exp\bigl((\theta-1)\log2\bigr)$, using $\exp(-\log2)=\frac12$; as $(\theta-1)\log2<0$, $0<r_{q}<1$. For $k$ a natural number or $0$, $\exp(-k\log2)=(\frac12)^{k}$ (with $(\frac12)^{0}=1$), and these powers are nonincreasing in $k$. For each natural number $m$ let

an interval with endpoints $(\frac12)^{m}\le(\frac12)^{m-1}$, which is Borel with $\mathcal{L}^{1}(A_{m})=(\frac12)^{m-1}-(\frac12)^{m}=(\frac12)^{m}$ by claim 4 of Existence of Lebesgue Measure on the Real Line ($\mathcal{L}^{1}$ is the measure $\lambda$ there, by Lebesgue Measure on $\mathbb{R}^n$). The $A_{m}$ are pairwise disjoint: if $m<m'$ and $s\in A_{m'}$, then $s\le(\frac12)^{m'-1}\le(\frac12)^{m}$, so $s\notin A_{m}$. For each natural number $n$ let $\bar h_{n}=\sum_{m=1}^{n}q^{m}\mathbf{1}_{A_{m}}$, which is measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, nonnegative, and nondecreasing in $n$; and let $\bar h(s)=\sup_{n}\bar h_{n}(s)$. By disjointness, $\bar h(s)=q^{m}$ if $s\in A_{m}$ and $\bar h(s)=0$ if $s$ lies in no $A_{m}$; so $\bar h:\mathbb{R}\to\mathbb{R}$ is nonnegative. By Monotone Convergence Theorem, $\bar h$ is measurable (hence Borel measurable, as in (a)) and $\int\bar h\,d\mathcal{L}^{1}=\sup_{n}\int\bar h_{n}\,d\mathcal{L}^{1}$; by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set, $\int\bar h_{n}\,d\mathcal{L}^{1}=\sum_{m=1}^{n}q^{m}(\frac12)^{m}=\sum_{m=1}^{n}r_{q}^{m}$, which by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates is at most $\sum_{m=1}^{\infty}r_{q}^{m}=r_{q}/(1-r_{q})$. Hence

Moreover, if $0<s<1$, then $\log s<0$, so $x_{s}=-\log s/\log2$ is positive, and by claim 4 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities $m=\lfloor x_{s}\rfloor+1$ is a natural number with $x_{s}<m\le x_{s}+1$; thus $-m\log2<\log s\le-(m-1)\log2$, and applying the increasing function $\exp$ gives $s\in A_{m}$. Hence

(c) The singular factor. Let $x\in W_{N}$, $(j,l)\in\Pi$ and $r\in\{1,2\}$, and write $s=s_{jl}$. We claim

If $s\ge1$, then $\log s\ge0$ and, by (3.1), $(\gamma-r)\log s\le\gamma\log s\le\gamma(\ell_{j}+\ell_{l})$. If $s<1$, take $m$ as in (6.2): if $\gamma\ge r$ then $(\gamma-r)\log s\le0\le\theta m\log2$; if $\gamma<r$ then $0<r-\gamma\le2-\gamma\le\theta$ and $(\gamma-r)\log s=(r-\gamma)(-\log s)\le\theta m\log2$; in both cases $\exp((\gamma-r)\log s)\le\exp(\theta m\log2)=q^{m}=\bar h(s)$. In either case (6.3) follows, both summands on its right being nonnegative.

(d) The Gibbs factor. For $x\in W_{N}$ let $E(x)=\exp(-P(x)/a)$ and $Q(x)=\exp\bigl(-\sum_{k}V(x_{k})/a+\gamma NL(x)\bigr)$. By The Logarithmic Energy of N Ordered Particles on the Weyl Chamber §energy and $b/a=\gamma$, $-P(x)/a=-\sum_{k}V(x_{k})/a+\gamma\sum_{(i,k)\in\Pi}\log s_{ik}$. By (3.1) and (3.2), $\sum_{(i,k)\in\Pi}\log s_{ik}\le(N-1)L(x)$, and the same bound holds for the sum over $\Pi$ with one pair $(j,l)$ removed, since $\ell_{j}+\ell_{l}\ge0$. Hence $E(x)\le Q(x)$. For $(j,l)\in\Pi$ and $r\in\{1,2\}$ we have $s_{jl}^{-r}=\exp(-r\log s_{jl})$, so