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Finite-spin Dobrushin comparison and boundary response

Statement

Assume the Axiom of Countable Choice. Let $d \geq 1$ , let $S$ be a finite nonempty set, and put $X = S^{Z^{d}}$ with its cylinder-generated Borel $σ$ -algebra. Let $γ$ and $\tilde{γ}$ be proper consistent finite-range specifications on $X$ , each with a uniform finite one-site influence radius. Finite range means that for every finite $Δ$ and cylinder event $A$ , each function $η \mapsto γ_{Δ} (1_{A} ∣ η)$ and $η \mapsto {\tilde{γ}}_{Δ} (1_{A} ∣ η)$ depends on only finitely many exterior coordinates, as in Lattice Gibbs kernels and DLR states. For each $i, j \in Z^{d}$ , let $C_{i i} = 0$ and let $C_{i j} \geq 0$ dominate both Dobrushin coefficients $c_{i j}^{γ}$ and $c_{i j}^{\tilde{γ}}$ of Finite-spin Dobrushin influence matrix. Assume $C_{i j} = 0$ whenever $∥ i - j ∥_{\infty} > R$ for some finite integer $R \geq 0$ and assume

$\sup_{i \in Z^{d}} \sum_{j \in Z^{d}} C_{i j} \leq c < 1.$

For each site $i$ , define the one-site conditional discrepancy

$b_{i} : = \sup_{η \in S^{Z^{d} ∖ {i}}} ∥ γ_{i} (\cdot ∣ η) - {\tilde{γ}}_{i} (\cdot ∣ η) ∥_{T V} .$

Write $C_{Λ} = (C_{i j})_{i, j \in Λ}$ and $D^{Λ} : = \sum_{n = 0}^{\infty} C_{Λ}^{n}$ for finite $Λ$ , with $C_{Λ}^{0}$ the identity matrix on $Λ$ ; also write $D_{i j} : = \sum_{n = 0}^{\infty} (C^{n})_{i j}$ for the infinite-lattice resolvent entries. These sums are finite entrywise, because every row sum of $C^{n}$ and $C_{Λ}^{n}$ is at most $c^{n}$ . A cylinder function is one depending only on the spins in some finite set. For a bounded real cylinder function $f : X \to R$ , define its oscillation by $δ_{i} (f) : = \sup_{σ, σ^{'} : σ_{k} = σ_{k}^{'} (k \neq i)} ∣ f (σ) - f (σ^{'}) ∣$ ; only finitely many $δ_{i} (f)$ are nonzero.

For every nonempty finite $Λ$ containing a finite coordinate set $K_{f}$ on which $f$ depends, and exterior configurations $ξ, ζ \in S^{Λ^{c}}$ , the corresponding finite-volume Gibbs distributions satisfy

$∣ γ_{Λ} (f ∣ ξ) - {\tilde{γ}}_{Λ} (f ∣ ζ) ∣ \leq \sum_{i \in Λ} δ_{i} (f) \sum_{k \in Λ} D_{i k}^{Λ} (b_{k} + \sum_{j \notin Λ} C_{k j} 1_{{ξ_{j} \neq ζ_{j}}}) .$

Moreover, each specification has a unique DLR state, denoted $μ$ and $\tilde{μ}$ , and every bounded real cylinder function satisfies

$∣ μ (f) - \tilde{μ} (f) ∣ \leq \sum_{i \in Z^{d}} δ_{i} (f) \sum_{j \in Z^{d}} D_{i j} b_{j} .$

In particular, when $γ = \tilde{γ}$ the comparison bound is zero. The estimates are conditional mathematical consequences of the stated finite-spin specifications and strict row bound.

Statement references

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Lattice Gibbs kernels and DLR states

Finite-spin Dobrushin influence matrix

Finite maximal coupling and heat-bath stationary law

Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence

Finite-range Dobrushin uniqueness and boundary influence

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges

Facts & Assumptions

Given: The finite-spin configuration space and DLR convention, two proper consistent finite-range specifications $γ, \tilde{γ}$ , a common finite-range matrix $C$ dominating both influence matrices, the strict row bound $c < 1$ , and a bounded cylinder function $f$ .

[A1]

Assume Countable Choice: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )). This is used only through the DLR existence theorem in [F6].

[F1]

A specification on $S^{Z^{d}}$ is proper and consistent when its kernels leave exterior events fixed and satisfy $γ_{Λ} γ_{Δ} = γ_{Λ}$ for finite $Δ \subseteq Λ$ ; its DLR states are probability measures satisfying $μ γ_{Λ} = μ$ for every finite $Λ$ . Finite range means that every finite-volume kernel on a cylinder event depends on finitely many exterior coordinates (Lattice Gibbs kernels and DLR states).

[F2]

The coefficient $c_{i j}^{γ}$ bounds the total-variation change in the one-site conditional law at $i$ when only boundary spin $j$ changes, with the same definition for $\tilde{γ}$ (Finite-spin Dobrushin influence matrix).

[F3]

For probability vectors on a finite set, a maximal coupling exists whose probability of disagreement equals their half-total-variation distance (Finite maximal coupling and heat-bath stationary law).

[F4]

Every bounded real sequence has a convergent subsequence (Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence).

[F5]

For $0 \leq c < 1$ , $\sum_{n = 0}^{\infty} c^{n} = 1 / (1 - c)$ and the tails tend to zero (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[F6]

Under Countable Choice, a proper consistent finite-alphabet specification with a uniform finite influence radius and strict Dobrushin row constant below $1$ has exactly one DLR state (Finite-range Dobrushin uniqueness and boundary influence).

Proof

1.1givenF1algebra

Fix finite $Λ$ and exterior configurations $ξ, ζ$ . Consistency in [F1] makes each finite-volume Gibbs distribution stationary under the random-scan heat-bath chain that chooses a site uniformly from $Λ$ and resamples its spin from the corresponding one-site kernel; the uniform mixture of invariant single-site updates remains invariant.

2.1F1F3F4step 1.1algebraconstruct

Couple the two finite-volume chains by choosing the same update site and using a maximal coupling from [F3] for the two one-site laws. Start the pair chain from the product of the two invariant marginals, so both marginals remain fixed at every time. The Cesàro averages of its distributions lie in a finite probability simplex; [F4] gives a convergent subsequence, and the limit is stationary because the difference between a Cesàro average and its one-step image is the initial-minus-final distribution divided by the averaging length. Its marginals are the two finite-volume Gibbs laws. Since a finite $Λ$ and finite range leave only finitely many exterior coordinates in these transition kernels, choose one such stationary law for each of the finitely many boundary-pattern pairs; this gives a measurable family for integrating over exterior configurations.

3.1F2F3step 2.1algebra

Let $q_{i}$ be the stationary probability that the two configurations disagree at $i \in Λ$ , and let $r_{i}$ be the mismatch probability if site $i$ is selected for update. One random-scan step leaves coordinate $i$ untouched with probability $1 - ∣ Λ ∣^{- 1}$ and updates it with probability $∣ Λ ∣^{- 1}$ ; stationarity therefore gives $q_{i} = (1 - ∣ Λ ∣^{- 1}) q_{i} + ∣ Λ ∣^{- 1} r_{i}$ , hence $q_{i} = r_{i}$ . The maximal-coupling property in [F3] makes $r_{i}$ the expected total-variation distance of the update laws. First change the kernel from $γ_{i}$ to ${\tilde{γ}}_{i}$ at the same boundary configuration, costing at most $b_{i}$ ; then change the differing internal and exterior spins one at a time. By [F2] and the envelope $C$ , the triangle inequality gives $q_{i} \leq b_{i} + \sum_{j \in Λ} C_{i j} q_{j} + \sum_{j \notin Λ} C_{i j} 1_{{ξ_{j} \neq ζ_{j}}}$ . The last sum is finite by the common finite-range assumption.

4.1F5step 3.1algebra

Put $e_{i} = \sum_{j \notin Λ} C_{i j} 1_{{ξ_{j} \neq ζ_{j}}}$ . Iterating the vector inequality in step 3.1 gives $q \leq \sum_{n = 0}^{N - 1} C_{Λ}^{n} (b + e) + C_{Λ}^{N} 1$ . Since every row sum of $C_{Λ}^{N}$ is at most $c^{N}$ , [F5] makes the last term tend to zero, yielding $q \leq D^{Λ} (b + e)$ . Coordinatewise telescoping of $f$ over its finite support then gives the stated finite-volume bound.

5.1F5step 4.1algebra

Let $Λ$ exhaust $Z^{d}$ . For fixed $i$ , each fixed-length path contributing to $(C_{Λ}^{n} b)_{i}$ is eventually contained in $Λ$ , and the omitted tail after length $N$ is at most $∥ b ∥_{\infty} c^{N + 1} / (1 - c)$ by [F5]. Thus $\sum_{n} (C_{Λ}^{n} b)_{i} \to (D b)_{i}$ . A path contributing to $(C_{Λ}^{n} e)_{i}$ must consist of $n$ internal edges followed by one edge to $Λ^{c}$ . Because each edge has length at most $R$ , every such path has at least $m_{Λ} (i) : = ⌊ {dist}_{\infty} (i, Λ^{c}) / R ⌋$ edges when $R > 0$ ; $m_{Λ} (i) \to \infty$ . The sum of weights of all paths of length at least $m_{Λ} (i)$ is at most $\sum_{m \geq m_{Λ} (i)} c^{m}$ , which tends to zero by [F5]. If $R = 0$ , every off-diagonal coefficient is zero and the boundary term is identically zero.

6.1A1F1F6step 4.1step 5.1∎

The hypotheses of [F6] apply separately to $γ$ and $\tilde{γ}$ , because their influence matrices are bounded entrywise by $C$ and hence have row constants at most $c < 1$ . Their unique DLR states exist. For each finite $Λ$ containing the support of $f$ , integrate the finite-volume estimate from step 4.1 against the product of the two exterior marginals of $μ$ and $\tilde{μ}$ . The DLR identities in [F1] identify the left side with $∣ μ (f) - \tilde{μ} (f) ∣$ ; then step 5.1 and the finite support of $f$ give the asserted infinite-volume inequality. Setting $b = 0$ gives equality of all cylinder expectations, hence equality of the two measures because cylinder events generate the Borel $σ$ -algebra.

Finite-spin Dobrushin comparison and boundary response

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources