Manuscript Oai227-A@1 · Oai227-A.tex · lemma · 7f45f607

Lemma: Dirichlet form and spin action

Stage 1: finite identities and the five-stage audit › Locally proved identities that an audit may use · lines 415–448 · proof included or linked

Oai227-A@1#lem:dirichlet, Oai227-A@1#eq:spin-action

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For real functions on the cube,

- ⟨ f, L_{J} f ⟩_{π} = i = 1 \sum^{n} π [Var_{π} (f ∣ σ_{- i})] .

Also,

L_{J} f_{i} = - f_{i} + tanh h_{i} .

The invariant law and both initialized processes are invariant under global spin reversal; in particular their one-spin means are zero.

Proof. Each $K_{i}$ is an orthogonal conditional-expectation projection on $L^{2} (π)$ . Hence

⟨ f, (I - K_{i}) f ⟩_{π} = π [f^{2}] - π [(K_{i} f)^{2}] = π [Var_{π} (f ∣ σ_{- i})] .

Only an update at site $i$ changes $f_{i}$ , and its conditional new mean is $tanh h_{i}$ , proving (eq:spin-action). Both the energy and the update rule respect global reversal. So do $π$ and $U_{n}$ .

Displayed formulas in this passage

Oai227-A@1#eq:spin-action · line 425
$L_{J} f_{i} = - f_{i} + tanh h_{i} .$
unlabeled · line 418
$- ⟨ f, L_{J} f ⟩_{π} = i = 1 \sum^{n} π [Var_{π} (f ∣ σ_{- i})] .$
unlabeled · line 437
$⟨ f, (I - K_{i}) f ⟩_{π} = π [f^{2}] - π [(K_{i} f)^{2}] = π [Var_{π} (f ∣ σ_{- i})] .$

Lemma: Dirichlet form and spin action

Displayed formulas in this passage

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