The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity

Fix parameters as in The Ising Population Data (i), adopt the Ising population data with those parameters, and adopt The Ising Population Model Instantiates the Data of the Fluctuation Theory and The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal, so that the equilibrium triple $(S,A,P)$ is a stationary mean-field triple for $\beta$, $(L,G)$ and the extensions $(U,V,\bar{\beta})$, $(U_{c},\bar{L},\bar{G})$. Let $H_{ij}(t)$ ($i,j\in\{1,2,3,4\}$) and $F_{\gamma\delta}$ ($\gamma,\delta\in\{1,2\}$) be the fluctuation Hessian coefficients of these data, and let $\mathcal{E}_{t}$, $\mathcal{B}_{t}$, $\tilde{\mathcal{E}}_{t}$, $H^{SS}_{t}$, $H^{SA}_{t}$, $H^{AS}_{t}$, $H^{AA}_{t}$, $F^{\star}$, $\Theta^{\star}_{t}$ and $\tilde{\Theta}^{\star}_{t}$ be the fluctuation LQG data of $(S,A,P)$ relative to $(U,V,\bar{\beta})$, $(U_{c},\bar{L},\bar{G})$ and the observation-rate extension $(\tilde{U},\bar{\tilde{\beta}})$. Following Completion of Squares and A Priori Control Bound for the Fluctuation Cost, define for $t\in[0,T]$ the real matrices $E_{t}$ ($2\times2$), $\mathsf{B}_{t}$ ($2\times2$), $Q_{t}$ ($2\times2$), $V_{t}$ ($2\times2$), $R_{t}$ ($2\times2$) and $\hat{F}$ ($2\times2$) by

$$E^{\delta\gamma}_{t}=\partial_{\gamma}\bar{b}^{\delta}(S_{t},A_{t}),\qquad \mathsf{B}^{\delta j}_{t}=\partial_{2+j}\bar{b}^{\delta}(S_{t},A_{t}),$$ $$Q^{\gamma\delta}_{t}=\tfrac{1}{4}\bigl(H_{\gamma\delta}(t)+H_{\delta\gamma}(t)\bigr),\quad V^{\gamma j}_{t}=\tfrac{1}{2}\bigl(H_{\gamma,2+j}(t)+H_{2+j,\gamma}(t)\bigr),\quad R^{ij}_{t}=\tfrac{1}{4}\bigl(H_{2+i,2+j}(t)+H_{2+j,2+i}(t)\bigr),\quad \hat{F}^{\gamma\delta}=\tfrac{1}{4}\bigl(F_{\gamma\delta}+F_{\delta\gamma}\bigr),$$

for $\gamma,\delta\in\{1,2\}$ and $i,j\in\{1,2\}$; these are the matrices written with the same symbols in that theorem, and the sans-serif $\mathsf{B}_{t}$ is distinct from the rate bound $B=\bar{a}$ while the matrices $V_{t}$ are distinct from the open set $V$ of clause The Ising Population Data (iii). Write $I$ for the $2\times2$ identity matrix. Then the following hold, for every $t\in[0,T]$.

1. (The Hessian coefficients.)(i) $F_{\gamma\delta}=0$ for all $\gamma,\delta$, and

$$H_{11}(t)=H_{22}(t)=\psi+2\mu,\qquad H_{12}(t)=H_{21}(t)=2\mu-\psi,\qquad H_{33}(t)=H_{44}(t)=\frac{1}{2\chi},$$

while $H_{ij}(t)=0$ for every other pair $(i,j)$. In particular $H_{ij}(t)=H_{ji}(t)$ for all $i,j$, and the coefficients do not depend on $t$.

2. (The coefficient matrices.)(ii)

$$E_{t}=-vv^{\top},\qquad \mathsf{B}_{t}=-\tfrac{1}{2}vv^{\top},\qquad Q_{t}=\tfrac{1}{2}\psi\,vv^{\top}+\mu\,\mathsf{n}\mathsf{n}^{\top},\qquad V_{t}=0,\qquad R_{t}=\frac{1}{4\chi}\,I,\qquad \hat{F}=0 .$$

3. (The fluctuation LQG data.)(iii)

$$\mathcal{E}_{t}=-vv^{\top}=E_{t},\qquad \mathcal{B}_{t}=-\tfrac{1}{2}vv^{\top}=\mathsf{B}_{t},\qquad \tilde{\mathcal{E}}_{t}=q\,I+q_{0}\,\mathsf{n}\mathsf{n}^{\top},$$ $$\Theta^{\star}_{t}=vv^{\top},\qquad \tilde{\Theta}^{\star}_{t}=\Bigl(\frac{q}{2}+q_{0}\Bigr)I,\qquad F^{\star}=0,$$

and the Hessian blocks are $H^{SS}_{t}=2Q_{t}$, $H^{SA}_{t}=0$, $H^{AS}_{t}=0$, $H^{AA}_{t}=(2\chi)^{-1}I$.

4. (Invertibility and the information matrix.)(iv) $R_{t}$ and $\tilde{\Theta}^{\star}_{t}$ are symmetric positive definite, hence invertible, with $R_{t}^{-1}=4\chi\,I$ and $(\tilde{\Theta}^{\star}_{t})^{-1}=\bigl(\tfrac{q}{2}+q_{0}\bigr)^{-1}I$. Putting $\tilde{D}_{t}=\tilde{\mathcal{E}}_{t}^{\top}(\tilde{\Theta}^{\star}_{t})^{-1}\tilde{\mathcal{E}}_{t}$ and

$$\tilde{d}=\frac{q^{2}}{\tfrac{q}{2}+q_{0}}=\frac{2q^{2}}{q+2q_{0}}>0 ,$$

we have

$$\tilde{D}_{t}=\frac{1}{\tfrac{q}{2}+q_{0}}\Bigl(q^{2}\,I+2q_{0}(q+q_{0})\,\mathsf{n}\mathsf{n}^{\top}\Bigr),\qquad \tilde{D}_{t}\,vv^{\top}=\tilde{d}\,vv^{\top},\qquad vv^{\top}\tilde{D}_{t}=\tilde{d}\,vv^{\top} .$$

5. (Hypothesis (JC) and hypothesis (H1).)(v) For every $w=(w^{1},w^{2},w^{3},w^{4})\in\mathbb{R}^{4}$,

$$\frac{1}{2}\sum_{i=1}^{4}\sum_{j=1}^{4}H_{ij}(t)\,w^{i}w^{j}=\frac{\psi}{2}\bigl(w^{1}-w^{2}\bigr)^{2}+\mu\bigl(w^{1}+w^{2}\bigr)^{2}+\frac{1}{4\chi}\Bigl(\bigl(w^{3}\bigr)^{2}+\bigl(w^{4}\bigr)^{2}\Bigr)\ \ge\ c_{J}\,|w|^{2},$$

where

$$c_{J}=\min\Bigl\{\psi,\ 2\mu,\ \frac{1}{4\chi}\Bigr\}>0 ,$$

and $\sum_{\gamma,\delta}F_{\gamma\delta}z^{\gamma}z^{\delta}=0\ge0$ for every $z\in\mathbb{R}^{2}$. Thus hypothesis (JC) of Localized Joint Coercivity of the Recentred N-Agent Cost Integrand under a Positive-Definite Fluctuation Hessian holds with the constant $c_{J}$, and hypothesis (H1) of Completion of Squares and A Priori Control Bound for the Fluctuation Cost holds with $r=c_{J}$, indeed with $r=(4\chi)^{-1}$.