Regimes of the Ising Population Model Outside the Fluctuation Optimality Theorem

This remark records what the hypotheses of The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States exclude in the model of The Ising Population Model: A Worked Instance of the Fluctuation Theory, and how its constants relate to the source. Nothing here is asserted as a result.

1. (The ferromagnetic sign.) In the source the interaction term of the running cost is $-\tfrac12\mathbf{J}(x^{2}-x^{1})^{2}$, and $\mathbf{J}>0$ makes congregation cheap; the present model has $\mathbf{J}=-\psi<0$, so it rewards splitting. The sign is not a matter of convenience. By The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §coercivity the quantity that hypothesis (JC) of Localized Joint Coercivity of the Recentred N-Agent Cost Integrand under a Positive-Definite Fluctuation Hessian requires to dominate $c_{J}|w|^{2}$ is

and with $\psi$ replaced by $-\mathbf{J}<0$ this is negative at $w=(1,-1,0,0)$, whatever $\mu$ and $\chi$ are. So (JC) fails for every positive constant in the ferromagnetic regime, and with it the standing hypotheses of the lower-bound theorem. Consider in particular the critical phase transition of the source, at $\chi\,\mathbf{J}=1$. Two facts about the present model, in which $\psi>0$ throughout, describe the drift of the fluctuation under the optimal feedback. First, the equation of The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form §scalar-control has the algebraic form $8\chi z^{2}+4z-\tfrac12\psi=0$ at a stationary point, whose roots are $(\pm\Delta-1)/(4\chi)$ with $\Delta=\sqrt{1+\chi\psi}$ as there. Second, clause The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §coefficients gives $E_{t}=-vv^{\top}$ and $\mathsf{B}_{t}=-\tfrac12 vv^{\top}$, clause The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §information gives $R_{t}^{-1}=4\chi I$, clause The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form §control-riccati gives $W_{t}=-z_{t}vv^{\top}$, and clause The Ising Population Data §dimensions gives $v\cdot v=2$, hence $vv^{\top}vv^{\top}=2vv^{\top}$. With these, the matrix obtained by substituting the optimal feedback of Completion of Squares and A Priori Control Bound for the Fluctuation Cost, namely the control $-R_{t}^{-1}W_{t}^{\top}$ applied to the state fluctuation, for the control in the linear fluctuation drift with state coefficient $E_{t}$ and control coefficient $\mathsf{B}_{t}$ is

whose coefficient $1+4\chi z_{t}$ equals $\Delta$ at the root $(\Delta-1)/(4\chi)$. Now substitute $\psi=-\mathbf{J}=-\chi^{-1}$ formally into these algebraic expressions — formally, because $\psi>0$ fails there, so neither lemma applies and there is no solution $z$ of the Riccati equation to speak of. Then $\Delta=0$: the two roots collapse to the double root $-(4\chi)^{-1}$, and the damping coefficient $1+4\chi z$ collapses with them to $0$, so that the mechanism which damps the fluctuation in the splitting regime is exactly the one that fails at the critical line while the noise continues to act. The critical line therefore lies outside the reach of the theorem as it stands. Making it accessible would require weakening (JC) to coercivity on the directions tangent to the simplex together with a separate treatment of the marginal case, not merely a different choice of extension.

2. (The convexity weight $\mu$.) Even in the splitting regime the $\psi$-term of the cost matrix — the summand $\tfrac12\psi\,vv^{\top}$ of $Q_{t}=\tfrac12\psi\,vv^{\top}+\mu\,\mathsf{n}\mathsf{n}^{\top}$ in clause The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §coefficients, equivalently the summand $\psi\,vv^{\top}$ of the Hessian block $H^{SS}_{t}=2Q_{t}$ of clause The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §lqg-data — is singular along $\mathsf{n}=(1,1)^{\top}$, the direction normal to the probability simplex, in which the state fluctuation cannot move. Hypothesis (JC) is nevertheless imposed on all of $\mathbb{R}^{l+m}$, so the extension of the running cost carries the term $\mu(x^{1}+x^{2}-1)^{2}$ of clause The Ising Population Data §cost. That term vanishes on the simplex, so it changes neither the running cost $L$ nor the control problem, and The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form §pairings records that it does not enter the value $V^{*}$; it only supplies curvature in a direction the model never uses. It is therefore an artifact of the form of (JC) rather than a feature of the model.

3. (The external field.) The source's external field $\mathbf{H}$ is taken to be $0$ here. For $\mathbf{H}\neq0$ the stationary point moves off the even split, the co-state no longer vanishes, and the Hessian coefficients pick up the contribution of the drift that the vanishing co-state suppresses in The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §hessian.

4. (Observations.) The background rate $q_{0}>0$ is needed for hypothesis (OC): with $q_{0}=0$ the aggregate observation drift of The Ising Population Model Instantiates the Data of the Fluctuation Theory §observations is $q\,x^{\upsilon}$, which vanishes at the two corners of the simplex, so no uniform positive lower bound exists there. At the other extreme, letting $q$ tend to $0$ with $q_{0}$ fixed drives $\tilde{d}=2q^{2}/(q+2q_{0})$ to $0$: the observation matrix becomes a multiple of $\mathsf{n}\mathsf{n}^{\top}$, which annihilates the tangential direction, the counts carry no information about the imbalance, and the filter Riccati equation of The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form §scalar-filter degenerates to the equation with $\tilde{d}=0$, whose equilibrium value is $\tfrac14$ — the covariance of the uncontrolled fluctuation. This is the degenerate case the source describes at $\mathbf{q}=0$.

5. (The entropic cost and the source's constants.) On $[\underline{a},\bar{a}]$ the function $\phi$ of The Regularised Entropic Rate Cost §cost-function satisfies, by The Regularised Entropic Rate Cost §profile, $\phi''(u)=u^{-1}$ with $\phi(1)=\phi'(1)=0$, so it agrees there with $u\mapsto u(\log u-1)+1$, the entropic rate cost of the source; the regularisation outside that interval only serves the global smoothness and bounded-derivative requirements of Twice Continuously Differentiable Extension of Population Cost Data, and the controls never leave $\mathcal{A}=[\underline{a},\bar{a}]^{2}$. The source computes in the reduced imbalance coordinate, so its scalars correspond to the action of the matrices used here on the tangential direction $v$: its $R=(4\boldsymbol{\beta})^{-1}$ and $V=0$ are the present $R_{t}=(4\chi)^{-1}I$ and $V_{t}=0$, its $E=-2$ is the eigenvalue of $\mathcal{E}_{t}=-vv^{\top}$ on $v$, and its stationary control Riccati root $(4\boldsymbol{\beta})^{-1}\bigl(-1+\sqrt{1-\boldsymbol{\beta}\mathbf{J}}\bigr)$ is the number $(\Delta-1)/(4\chi)$ of The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form §scalar-control, read with $\boldsymbol{\beta}=\chi$ and $\mathbf{J}=-\psi$. Constants attached to the state noise and to the filter covariance differ from the source's by the factors implicit in that reduction.