2. The D-metric

The exposition of this section follows [Reference MacKay13] but fills some gaps.

Let X be the product of a set of metric spaces $(X_s, d_s)$ , for ‘sites’ s in a countable set S (countable includes finite). The spaces $(X_s, d_s)$ are assumed to be Polish (complete separable metric spaces) with bounded diameter, $\Omega = \sup_{s\in S} \mbox{diam}_s(X_s) < \infty$ , and to have at least two points each. The product X is endowed with product topology and the resulting Borel sets. By a measure I mean a finite signed real Borel measure.

With the above assumptions, X is compact and the space $\mathcal{M}$ of measures is the dual of the space $C(X,\mathbb{R})$ of continuous functions from X to $\mathbb{R}$ with supremum norm $|\ |_\infty$ . Given a measure m and a continuous function $f\colon X \to \mathbb{R}$ , m(f) (or just mf) denotes the integral of f with respect to m. Define $|m|_1 = \sup \{m(f); f \in C(X,\mathbb{R}), |f|_\infty \le 1\}$ .

The set $\mathcal{P}$ of probabilities on X consists of the measures p satisfying $p(X)=1$ and $p(Y)\ge 0$ for all Borel subsets Y. In particular, $p \in \mathcal{P}$ implies $p(\textbf{1})=1$ , where $\textbf{1}$ is the function taking the value 1 everywhere on X. Also, $|p|_1 = 1$ .

Definition 1. For a function $f \colon X \to \mathbb{R}$ , its Lipschitz constant with respect to variations on site $s\in S$ is

\begin{align*}\Delta_s(f) = \sup \frac{f(x)-f(x')}{d_s(x_s,x'_s)}\end{align*}

over $x,x' \in X$ differing at site s and agreeing elsewhere.

Note that $\Delta_s(f)\ge 0$ and may be $+\infty$ .

Note that Dobrushin smooth functions are automatically continuous: firstly,

\begin{align*}|f(x)-f(y)| \le \sum_s \Delta_s(f)\, d_s(x_s,y_s);\end{align*}

secondly, given $\varepsilon>0$ , there exists a finite (possibly empty) subset $K\subset S$ such that $\sum_{s\in S\setminus K} \Delta_s(f) <\varepsilon/\Omega$ and $\Delta_s(f)>0$ for $s \in K$ . For $s \in K$ , let $A_s \subset X_s$ be the open ball of radius $\varepsilon /(|K|\Delta_s(f))$ about $y_s$ , and for $s \in S\setminus K$ , let $A_s = X_s$ . Then $\prod_{s\in S} A_s$ is an open neighbourhood of y in X, and $x \in \prod_{s\in S} A_s$ implies $|f(x)-f(y)| < \varepsilon$ .

Also, letting C be the set of constant functions, then $|\cdot |_F$ is a norm on the quotient space $F/C$ (F modulo addition of constants).

It is a norm on Z; see [Reference MacKay13]. With this norm, the dual space $Z^*$ is $F/C$ (proved in Appendix A).

The D-metric makes ${\mathcal{P}}$ into a complete metric space, with diameter $\Omega=\sup_{s\in S} \mbox{diam}_s(X_s)$ . The proof of completeness in [Reference MacKay13] is wrong (the direction of duality above was mistaken), but a proof is provided here in Appendix B.

It is a norm on $\mathcal{M}$ , and satisfies $|m|_\mathcal{M} \le \Omega |m|_1$ . Note that for $\mu \in Z$ , $|\mu|_\mathcal{M} = |\mu|_Z$ , because if $|f|_F \le 1$ , we can add a constant to f to achieve $|f|_\infty \le \Omega$ and it does not change $\mu(f)$ .

Note that a transition operator maps Z to Z (write $\mu \in Z$ as $\mu^+-\mu^-$ with $\mu^\pm$ non-negative; $\mu \textbf{1} = 0$ so $\mu^\pm \textbf{1}$ are equal; if $\mu \ne 0$ then the common value $k>0$ ; then $\mu^\pm/k \in \mathcal{P}$ and it follows that $(\mu T) \textbf{1} = 0$ ). Furthermore, to check T is bounded it is enough to check that its restriction $T_Z\colon Z \to Z$ is bounded. This is because, choosing a $p \in \mathcal{P}$ , any $m \in \mathcal{M}$ can be written as $kp + \mu$ for some $k \in \mathbb{R}$ , $\mu \in Z$ (simply let $k = m\textbf{1}$ and $\mu = m-kp$ ). Then, for $|f|_F \le 1$ , $|f|_\infty \le \Omega$ ,

\begin{align*}(mT)f = k(p T)f + (\mu T)f \le |k||f|_\infty + |\mu T|_Z |f|_F \le k\Omega + |\mu|_Z |T_Z|_Z.\end{align*}

It is a complete norm on F.

A transition operator T has a dual acting on F that we denote by the same symbol but acting to the right.

Proof. First, prove the uniqueness of Tf. If there are $g_1,g_2 \in F$ such that, for all $p \in \mathcal{M}$ , $p(g_1)=(pT)f=p(g_2)$ , let $g = g_1-g_2$ . Then, for all $\mu\in Z$ , $\mu(g) = 0$ , which implies that g is constant. Then, choosing a $\pi \in \mathcal{P}$ , $\pi(\textbf{1})=1$ , so $\pi(g)=0$ implies $g=0$ .

Next, prove the existence of Tf. Given $x \in X$ , let $g(x) = (\delta_x T)f$ (recall that $\delta_x$ is the atomic probability at x). We have to check that the function g is in F. Making a change in x at a single site s, we obtain $g(x')-g(x) \le |\delta_{x'}-\delta_x|_Z |T|_Z |f|_F$ . Noting that $|\delta_{x'}-\delta_x|_Z = d_s(x'_s,x_s)$ , we obtain $\Delta_s(g) \le |T|_Z |f|_F$ . To bound the sum of the $\Delta_s(g)$ , let $\gamma \in (0,1)$ . For each $s \in S$ , let $x^{(s)}, y^{(s)}$ be points of X differing only at site s and that achieve $g(x^{(s)})-g(y^{(s)}) \ge \gamma \Delta_s(g) d_s(x^{(s)},y^{(s)})$ . Let $\mu \in Z$ be the sum of ‘dipoles’

\begin{align*}\frac{\delta_{x^{(s)}}-\delta_{y^{(s)}}}{d_s(x^{(s)},y^{(s)})}\end{align*}

over a finite subset $S'\subset S$ . Then $|\mu|_Z=1$ and

\begin{align*}\gamma \sum_{s\in S'} \Delta_s(g) \le \mu(g) = (\mu T)f \le |\mu T|_Z |f|_F \le |T|_Z |f|_F.\end{align*}

This is true for all finite $S'\subset S$ , hence $\gamma\sum_{s\in S}\Delta_s(g) \le |T|_Z|f|_F$ . So $g \in F$ .

Elaborating on the proof, we can obtain useful bounds. Specifically, taking $\gamma\to 1$ we obtain $|Tf|_F \le |T|_Z |f|_F$ . Also, $|Tf|_\infty = \sup_x |(\delta_xT)f | \le |f|_\infty$ . We quantify the size of a transition operator T by its operator norm on F:

(1) \begin{equation}\|T\| = \sup \{ \|Tf\| ; \|f\| \le 1 \}.\end{equation}

Note that $\|T\| \le \max\!(|T|_Z,1)$ .

It will be convenient in much of this paper to consider the action of T on F rather than $\mathcal{M}$ , because F is complete. The property that T preserves probability can be written as $T\textbf{1} =\textbf{1}$ .

The reason for introducing the Dobrushin metric and the concept of smoothness in [Reference MacKay13] was to derive the following theorem.

A different norm on transition operators was used in [Reference MacKay13], but it is equivalent to the operator norm (1). Also, it was mistakenly assumed in [Reference MacKay13] that Z is complete and hence that if a bounded operator on Z is invertible then its inverse is bounded; it is not clear that this always holds for $(I-T)_Z$ so we include the requirement for its inverse to be bounded in the statement of the theorem. With this addition, the proof in [Reference MacKay13] goes through. There are sufficient conditions for bounded invertibility of $(I-T)_Z$ in terms of Dobrushin’s ‘dependency matrix’ [Reference Dobrushin7], which are verifiable in relevant classes of system, so this is a useable result [Reference MacKay13].

3. Spectral projections

We begin the discussion of spectral projections in the general context of bounded linear operators on Banach spaces.

Because the two parts of the spectrum are closed and bounded, it follows that the distance between them is positive, called a spectral gap.

An example of a spectral projection is $P= \textbf{1} p$ for the stationary probability p for a geometrically ergodic transition operator T. This is because $p \textbf{1} = 1$ for any probability p, $T \textbf{1} = \textbf{1}$ by the definition of a transition operator, $p T = p$ for a stationary measure, and geometric ergodicity implies the eigenspace for eigenvalue 1 is one-dimensional and the rest of the spectrum is in a disk of radius less than 1 around 0. Theorem 1 gives smooth dependence of p on T. Using the norm (1) on spectral projections we find that $\|\textbf{1} p'\| = |p'|_Z / \Omega$ , so we see that Theorem 1 gives smooth dependence of the spectral projection $\textbf{1} p$ on T. The goal of this section is to extend this result to arbitrary spectral projections for T.

Returning to the general context of bounded linear operators on a Banach space F, the set M of (bounded) projections on F is a smooth manifold (a variant of a Grassmann manifold). One way to see this is that projections are equivalent to direct sum decompositions $R\oplus K$ of F into two closed subspaces, namely the projection onto R along K. The space of closed subspaces is the Grassmann manifold. The space of complementary pairs of closed subspaces is also a manifold.

Here is an outline proof that M is a manifold, from the definition as the set of projections, because the ingredients are useful later. Let B(F) be the space of bounded linear operators P on the space F, with operator norm, which makes B(F) a Banach space. Define $\Phi$ on B(F) by $\Phi(P) = P^2-P$ , so $M= \Phi^{-1}(0)$ . Given ${P_0}\in M$ , let $R = \mbox{im}\,{P_0}$ and $K=\ker {P_0}$ . Both are closed. Then $F = R \oplus K$ . Relative to this direct-sum decomposition,

(2) \begin{equation} {P_0} = \Bigg[\begin{array}{c@{\quad}c} I & 0 \\[5pt] 0 & 0 \end{array}\Bigg].\end{equation}

For an arbitrary operator

\begin{equation*} \pi = \Bigg[\begin{array}{c@{\quad}c} \pi_1 & \pi_2 \\[5pt] \pi_3 & \pi_4 \end{array} \Bigg]\ \mathrm{ on }\; F, \quad D\Phi_{{P_0}} (\pi) = \Bigg[\begin{array}{c@{\quad}c} \pi_1 & 0 \\[5pt] 0 & -\pi_4 \end{array} \Bigg].\end{equation*}

With respect to the same direct-sum decomposition, let $T_{{P_0}}M$ (note the reuse of the symbol T, which will signify tangent space instead of transition operator) be the space of operators on F of the form

(3) \begin{equation} \Bigg[\begin{array}{c@{\quad}c} 0 & \pi_2 \\[5pt] \pi_3 & 0 \end{array}\Bigg],\end{equation}

and $N_{{P_0}}M$ be those of the form

\begin{align*}\Bigg[\begin{array}{c@{\quad}c} \pi_1 & 0 \\[5pt] 0 & \pi_4 \end{array} \Bigg].\end{align*}

Consequently, $D\Phi_{{P_0}}$ maps $N_{{P_0}}M$ to itself by $(\pi_1,\pi_4) \mapsto (\pi_1,-\pi_4)$ , which has bounded inverse, namely itself. So the implicit function theorem shows that the set of P for which the diagonal blocks of $P^2-P$ in the decomposition $R\oplus K$ are zero is locally the graph of a $C^1$ function $\psi\colon T_{P_0}M \to N_{P_0}M$ .

To complete the outline proof, we have to check that the off-diagonal blocks of $P^2-P$ automatically evaluate to zero too. The way we found to do this is via an explicit formula for $\psi$ , as follows. Writing $P=P_0+\pi$ , the equations for the diagonal blocks of $P^2-P$ to be zero are $\pi_1+\pi_1^2 = -\pi_2\pi_3$ , $\pi_4-\pi_4^2 = \pi_3\pi_2$ . The unique solution $\psi$ for $\pi$ small is

\begin{align*}\pi_1 = -\tfrac12 I + \sqrt{\tfrac14 I - \pi_2\pi_3}, \qquad \pi_4 = \tfrac12 I - \sqrt{\tfrac14 I - \pi_3\pi_2},\end{align*}

where the square roots are defined by their binomial expansions. Then the off-diagonal blocks $\pi_1\pi_2+\pi_2\pi_4$ , $\pi_3\pi_1 + \pi_4 \pi_3$ can be seen to evaluate to 0. Note that this explicit formula for $\psi$ gives a direct proof that M is a manifold, rendering use of the implicit function theorem redundant. Perhaps there is a simpler way to show the off-diagonal blocks are zero.

Note that $\Phi(P_0+\tilde{\pi} + \psi(\tilde{\pi}))=0$ for $\tilde{\pi} \in T_{P_0}M$ near 0 (or the explicit formula for $\psi$ ) implies $D\psi{(0)} = 0$ , so $T_{P_0}M$ is the tangent space to M at ${P_0}$ (hence the notation). Note also that the manifold M of projections has many components of different dimensions, corresponding to the similarity class (rank in finite dimensions) of the projection.

Then we have the following classical theorem. It is often proved by contour integration of the resolvent operator, e.g. [Reference Kato10], but we prefer our own proof.

Proof. Define a map $G\colon M \times B(F) \to B(F)$ by the commutator $G(P,T) = [P,T] = PT-TP$ . Note that the image lies in $T_PM$ , because in the direct sum decomposition $F=R\oplus K$ above, where P has the form (2), then [P, T] has the form (3). Thus G can be considered as a map from B(F) to sections of TM.

Now we apply the implicit function theorem to G. We have $G(P_0,T_0)=0$ , and G is $C^r$ . It remains to check that the derivative ${\partial G}/{\partial P}$ at $(P_0,T_0)$ , mapping $T_{P_0}M$ to itself, is invertible with bounded inverse. We denote it for short by L.

Applied to a tangent vector $P' \in T_PM$ , $L(P') = P'T-TP' \in T_PM$ . In the direct sum decomposition $F=R \oplus K$ ,

\begin{align*} T \;\mathrm{ has\; the\; form\ } \Bigg[\begin{array}{c@{\quad}c} T_P & 0 \\[5pt] 0 & T_Q\end{array}\Bigg]\ \mathrm{ and }\; P'\; \mathrm{ has\; the\; form\ } \Bigg[\begin{array}{c@{\quad}c} 0 & U \\[5pt] V & 0\end{array}\Bigg]. \end{align*}

So

\begin{equation*} L(P') = \Bigg[ \begin{array}{c@{\quad}c} 0 & UT_Q-T_PU \\[5pt] VT_P - T_Q V & 0 \end{array} \Bigg]. \end{equation*}

A general element of $T_PM$ has the form

\begin{align*}\Bigg[\begin{array}{c@{\quad}c} 0 & C \\[5pt] D & 0 \end{array}\Bigg].\end{align*}

We claim that L considered as mapping $T_PM$ to itself is invertible, because inverting L reduces to solving two ‘Sylvester equations’ $UT_Q-T_PU = C$ , $VT_P-T_QV = D$ , for operators U, V. There is a unique bounded solution to each of the Sylvester equations if and only if the spectra of $T_P$ and $T_Q$ are disjoint [Reference Bhatia and Rosenthal4]. It is automatic that the resulting inverse operators $C \mapsto U, D \mapsto V$ are bounded, because the inverse of an invertible bounded linear operator between Banach spaces is bounded [Reference Kato10].

Thus, L has bounded inverse and the result follows by the implicit function theorem.

Note that the result does not preclude existence of other spectral projections further away from $P_0$ , for example of different rank.

Denoting derivatives with respect to $\mu$ by ^′, we note the useful formula $P' = -L^{-1} [P,T']$ . This results from applying the chain rule to the equation $G(P_\mu, T_\mu) = 0$ at $\mu=0$ , but holds more generally if L is evaluated at $(P_\mu,T_\mu)$ . If we have bounds on $L^{-1}$ and T ^′, it allows us to deduce bounds on $P_\mu-P_0$ .

We now apply Theorem 2 to the case of F being the Dobrushin smooth functions on a countable product of Polish spaces with bounded diameter, with the norm of Definition 8 and the associated norm (1) on linear operators on F, such as transition operators T, projections P, and infinitesimal changes to each, and smoothness as in Definition 9.

For practical purposes, we need a bound on the solutions of the Sylvester equations.

Definition 11. For bounded linear maps $A\colon K \to K$ , $B\colon H \to H$ between normed spaces, their separation is

\begin{align*}\mbox{sep}(A,B) = \inf_{X\ne 0} \frac{\|AX-XB\|}{\|X\|}\end{align*}

over bounded linear $X\colon H\to K$ .

This generalises the concept of the separation of two matrices A, B, which is usually defined using the Frobenius norm [Reference Stewart20, Reference Varah21]. Note that $\mbox{sep}(B,A)$ is not necessarily equal to $\mbox{sep}(A,B)$ . From the definition, $\mbox{sep}(A,B) >0$ if and only if the Sylvester equation $AX-XB = C$ has a unique solution X for each C, and by the theory of Sylvester equations already used, this is true if and only if the spectra of A and B are disjoint. Then $\| X \| \le \| C \| /\mbox{sep}(A,B)$ . Thus, taking $A=T_P$ and $B=T_Q$ ,

\begin{align*}\|L^{-1}\| \le \gamma = (\mbox{sep}(A,B))^{-1}+(\mbox{sep}(B,A))^{-1}.\end{align*}

In some cases we can give explicit lower bounds on $\mbox{sep}(A,B)$ . For example, if the spectral radii $\rho_A, \rho_{B^{-1}}$ of A and $B^{-1}$ satisfy $\rho_A \rho_{B^{-1}} < 1$ then for $\lambda>1$ there exist $C_A,C_{B^{-1}}$ (dependent on $\lambda$ ) such that, for $n\ge 0$ , $\|A^n\| \le C_A (\lambda\rho_A)^n$ and $\|B^{-n}\| \le C_{B^{-1}} (\lambda\rho_{B^{-1}})^n$ . The explicit (convergent) solution $X=-\sum_{n=0}^\infty A^n C B^{-n-1}$ of the Sylvester equation shows that

\begin{align*}\mbox{sep}(A,B) \ge \frac{1-\lambda^2 \rho_A \rho_{B^{-1}}}{C_A C_{B^{-1}} \lambda \rho_{B^{-1}}}.\end{align*}

Similarly, for $YA-BY=D$ the explicit solution $Y=-\sum_{n\ge 0} B^{-n-1}DA^n$ provides the same bound on $\mbox{sep}(B,A)$ . A method to obtain similar bounds for some more general forms of separation of the spectra is described in [Reference Nevanlinna16].

The important point is that for a family of examples with growing system size N, if the separation is bounded away from zero then the continuation in Theorem 3 is uniform in N. Furthermore, it can apply to infinite systems.

The same can be done in continuous time, with the transition operator replaced by a transition generator, but we do not spell it out here, save to mention that if the spectra of A and B lie in ${\mathrm{Re}\,} z \ge r_A$ , ${\mathrm{Re}\,} z \le r_B$ respectively, with $r_A>r_B$ , the explicit (convergent) solution $X=\int_0^\infty {\mathrm{e}}^{(-A+r)t}C{\mathrm{e}}^{(B-r)t}\,{\mathrm{d}} t$ for $r \in (r_B,r_A)$ provides a bound of the form

\begin{align*}\mbox{sep}(A,B) \ge \frac{r_A-r_B-2\varepsilon}{c_A c_B}\end{align*}

for any $\varepsilon>0$ , with $c_A,c_B$ depending on $\varepsilon$ .

4. An illustration

As an example, consider a three-state PCA, with local state space $\{+,0,-\}$ at each site of a finite undirected graph with N nodes and bounded degree, say by m. The transition probabilities for the state at a given site are taken to be

(4) \begin{equation} \left[ \begin{array}{c@{\quad}c@{\quad}c} 1-(1+\alpha n_- )\varepsilon & (1+\alpha n_-)\varepsilon & 0 \\[5pt] \frac12 & 0 & \frac12 \\[5pt] 0 & (1+\alpha n_+)\varepsilon & 1- (1+\alpha n_+)\varepsilon \end{array} \right] \end{equation}

in basis $(+,0,-)$ , where $n_\pm$ are the numbers of neighbours in states $\pm$ respectively, $\alpha \ge 0$ , and $\varepsilon \in [0,(1+\alpha m)^{-1}]$ . For $\varepsilon=0$ the system has eigenvalue 1 with multiplicity $2^N$ and eigenvalue 0 with multiplicity $3^N-2^N$ .

From the above theory, the spectral projection to the subspace for eigenvalue 1 has a continuation for $\varepsilon$ small. The continuation is uniform in N because the separation of the relevant operators can be bounded away from 0 uniformly in N. The dynamics on the image of the projection is slow because the spectrum moves from $\{1\}$ by at most $O(\varepsilon)$ . It is still Markovian. It might loosely be considered as an effective PCA with two states $\{+,-\}$ on each site but $R=\mbox{im}\, P$ is not spanned by $\delta$ -functions on states, so such a description requires interpretation analogous to quasiparticles in quantum mechanics.

If $\alpha < 1/m$ and $\varepsilon>0$ , the dynamics on R is geometrically ergodic, because the whole system is. The latter can be proved using Dobrushin’s dependency matrix, as follows (compare an example of its use in [Reference MacKay13]). Take the discrete metric on the local state spaces. Then the update probability distributions $p(\sigma)$ for the state at a site given its current state $\sigma$ and the numbers $n_\pm$ are the rows of the matrix (4) and so the variation distance for a change of state on the given site is at most $(1-\varepsilon)$ , and for a change on a neighbouring site is at most $\alpha \varepsilon$ . Thus the dependency matrix has $\ell_\infty$ -norm at most $1-(1-m\alpha)\varepsilon$ . If $\alpha < 1/m$ and $\varepsilon>0$ , this is less than 1 and so the system is geometrically ergodic.

5. Second-order perturbation theory

We might ask what the dynamics of the above example look like on the image of the projection corresponding to the spectrum near 1. This can be answered by analogy to second-order perturbation theory computations in quantum mechanics, e.g. the derivation of the $t-J$ model from the Hubbard model [Reference Spalek17]. In the above example, the 0 state mediates interactions between the $\pm$ states. Here, a general treatment of second-order perturbation theory for families of stochastic operators is given and applied to this example.

Under the spectral gap condition at $\varepsilon=0$ and with the norm (1), it has been proved above that for a smooth family of stochastic operators $T_\varepsilon$ and a spectral projection $P_0$ for $T_0$ , there is an open neighbourhood in $\varepsilon$ for which $P_0$ continues smoothly to a spectral projection $P_\varepsilon$ for $T_\varepsilon$ . It is convenient to write $P_\varepsilon = \psi_\varepsilon P_0 {\psi_\varepsilon}^{-1}$ for an $\varepsilon$ -dependent invertible bounded linear map $\psi$ , with $\psi_0$ the identity. There is a lot of freedom in the choice of $\psi$ , the only constraints being that $\psi_\varepsilon(\mbox{im}\,P_\varepsilon) = \mbox{im}\,P_0$ and $\psi_\varepsilon(\ker P_\varepsilon)= \ker P_0$ , but it is good also to take $\psi$ to be as smooth in $\varepsilon$ as is T. Then we can define $S = \psi' \psi^{-1}$ , where ^′ denotes ${{\mathrm{d}}}/{{\mathrm{d}}\varepsilon}$ , and the above conditions reduce to

(5) \begin{equation}[S,P]=P',\end{equation}

where $[\,\cdot\,,\cdot\,]$ again denotes the commutator. A convenient solution is $S = P'(P-Q)$ , which can be checked to satisfy the condition (5).

Then it is desired to compute $\hat{T} = \psi^{-1}T\psi$ on $\mbox{im}\,P_0$ . This is the stochastic operator that represents the dynamics of $T_\varepsilon$ on $\mbox{im}\,P_\varepsilon$ , using the coordinate system $\psi_\varepsilon$ .

Start from $\hat{T}_0 = T_0$ . The first derivative is $\hat{T}' = \psi^{-1}(T'+[T,S])\psi$ . Using the above choice of S, we obtain $\hat{T}' = \psi^{-1}J\psi$ , where $J = PT'P+QT'Q$ . The second derivative can be evaluated to $\hat{T}'' = \psi^{-1}(PT''P+QT''Q + [[T,P'],P'])\psi$ .

Evaluating these at $\varepsilon=0$ , we obtain $\hat{T}$ to second order in $\varepsilon$ as

(6) \begin{equation}\hat{T}_\varepsilon = P_0 T_\varepsilon P_0 + Q_0 T_\varepsilon Q_0 + \frac{\varepsilon^2}{2}[[T_0,P'_{\!\!0}],P'_{\!\!0}].\end{equation}

It is perhaps more useful to substitute $[T,P'] = [T',P]$ , since the right-hand side of this is readily computable, but the second occurrence of P ^′ above means that solution of this equation for P ^′ is required.

For the example of Section 4, there is already an effect at first order in $\varepsilon$ , so second-order perturbation theory is perhaps unnecessary, but it still serves as an illustration of the procedure.

\begin{align*}T_0 = P_0 = \otimes_{s\in S} \left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0 \\[5pt] \frac12 & 0 & \frac12 \\[5pt] 0 & 0 & 1 \end{array} \right], \qquad T'_{\!\!0} = \otimes_{s\in S} \left[\begin{array}{c@{\quad}c@{\quad}c} -\beta_- & \beta_- & 0\\[5pt] 0 & 0 & 0 \\[5pt] 0 & \beta_+ & -\beta_+ \end{array} \right],\end{align*}

where $\beta_\pm = 1+ \alpha n_\pm$ . Thus

\begin{align*}P_0 T_\varepsilon P_0 + Q_0 T_\varepsilon Q_0 = \otimes_{s\in S} \left[\begin{array}{c@{\quad}c@{\quad}c} 1-\beta_-\varepsilon/2 & 0 & \beta_-\varepsilon/2 \\[5pt] \tfrac12 + \beta_+\varepsilon/2 & (\beta_++\beta_-)\varepsilon/2 & \tfrac12 + \beta_-\varepsilon/2 \\[5pt] \beta_+\varepsilon/2 & 0 & 1-\beta_+\varepsilon/2 \end{array}\right].\end{align*}

To compute the second-order term, begin with

\begin{align*}[T'_{\!\!0},P_0] = \otimes_{s \in S} \left[\begin{array}{c@{\quad}c@{\quad}c} \frac12 \beta_- &-\beta_- &\frac12 \beta_- \\[5pt] \frac12\beta_- &-\frac12 (\beta_-+\beta_+) & \frac12 \beta_+ \\[5pt] \frac12 \beta_+ & -\beta_+ & \frac12 \beta_+ \end{array} \right].\end{align*}

Using $P_0P'_{\!\!0}=P'_{\!\!0}Q_0$ , $P'_{\!\!0}$ has just four independent parameters on each site, and solving $[T,P']=[T',P]$ for them yields

\begin{align*}P'_{\!\!0} =\otimes_{s\in S} \left[ \begin{array}{c@{\quad}c@{\quad}c} \frac12 \beta_- & -\beta_- & \frac12 \beta_- \\[5pt] \frac12 \beta_+ & -\frac12 (\beta_-+\beta_+) & \frac12 \beta_- \\[5pt] \frac12 \beta_+ & -\beta_+ & \frac12 \beta_+ \end{array} \right].\end{align*}

To compute the $\varepsilon^2$ term in (6), we can first subtract $[T'_{\!\!0},P_0]$ from $P'_{\!\!0}$ , leaving, site-wise,

\begin{align*}[[T_0,P'_{\!\!0}],P'_{\!\!0}] = \frac{\beta_+-\beta_-}{2}\left[\![T'_{\!\!0},P_0],\! \left[\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & 0 \\[5pt] 1 & 0 & -1\\[5pt] 0 & 0 & 0\end{array}\right]\right] =\frac{\beta_+-\beta_-}{2} \!\left[\begin{array}{c@{\quad}c@{\quad}c} -\beta_- & 0 & \beta_-\\[5pt] -\beta_- & \beta_- - \beta_+ & \beta_+ \\[5pt] -\beta_+ & 0 & \beta_+ \end{array}\right]\!.\end{align*}

Thus, the final result to second order, using the $(+,-)$ part of the $\psi_\varepsilon$ basis, is the effective PCA

\begin{align*}\hat{T}_\varepsilon = \otimes_{s\in S} \Bigg[\begin{array}{c@{\quad}c} 1-\varepsilon\beta_-/2+\varepsilon^2\beta_-(\beta_- -\beta_+)/4 & \varepsilon\beta_-/2-\varepsilon^2\beta_-(\beta_- -\beta_+)/4 \\[5pt] \varepsilon \beta_+/2-\varepsilon^2\beta_+(\beta_+ -\beta_-)/4 & 1-\varepsilon \beta_+/2 + \varepsilon^2\beta_+(\beta_+ -\beta_-)/4 \end{array} \Bigg].\end{align*}

The outcome is a two-state model in which to leading order there is a small probability $\varepsilon \beta_\mp/2 = \varepsilon (1+\alpha n_\mp)/2$ per timestep for transition from state $+$ to $-$ , respectively $-$ to $+$ .

Perhaps a reader can suggest (and apply the method to) a more significant example.

6. Further potential applications

One area to which the above results might be usefully applied is metastability. Metastability is the phenomenon that an ergodic process may spend long times exploring restricted subsets of the support of the probability distribution, switching between them infrequently but sufficiently to achieve ergodicity in the long run. For an infinite limit, there may be more than one stationary distribution. One reference is [Reference Bovier and den Hollander5].

As an application of the result, we can start from [Reference Davies6] in which the hypothesis is a reversible Markov process in continuous time with spectrum in $[\!-\!\varepsilon,0] \cup [\!-\!\infty,-1]$ such that each function in the image of the spectral projection P for $[\!-\!\varepsilon,0]$ is bounded, and it is proved that there is a partition of the state spaces into ‘metastable regions’. It is not clear to me, however, whether there are relevant examples satisfying the hypotheses. We might think that Glauber dynamics of the 2D Ising model below the critical temperature would qualify, but I am not aware that it is proved to have a spectral gap. Nevertheless, if there are examples on product spaces then the result of the present paper proves robustness of spectral gap and hence of the phenomenon of metastability.

We could envisage the result also being useful to treat perturbation of Markov dynamics with more than one stationary distribution, for example with more than one communicating component. Addition of some interaction between the communicating components typically reduces the system to a single communicating component, but the result of this paper shows there is a continuation of the spectral projection to a spectral projection with spectrum contained near 1 and the same rank (equal to the original number of communicating components), and it gives strong control over the resulting continuation. More substantially, we would like to deduce something about metastability in ergodic finite versions of infinite PCA with non-unique stationary distribution.

Another possible application is to perturbations of product systems in which the units all have simple eigenvalue +1 and isolated spectrum near some $\lambda$ in the open unit disk. The conclusion is that there persists an invariant subspace with decay constant near $\lambda$ .