Manuscript Oai265-A@2 · Oai265-A.tex · lemma · c0a34a45

Lemma: Square-lattice boundary accounting

Stage 0: the frozen finite-domain package › Boundary conversions · lines 315–346 · proof included or linked

Oai265-A@2#lem:boundary

For every $A \subseteq Λ$ ,

∣ \partial_{in} A ∣ \leq ∣ \partial_{Λ} A ∣ \leq 4∣ \partial_{in} A ∣,

and

∣ \partial_{Λ} A ∣ \leq 2∣ Z_{A} ∣.

Consequently, if an individual state has an independently established bound (eq:AL), then

S_{Ω} (A) \leq 4 C ∣ \partial_{in} A ∣, S_{Ω} (A) \leq 2 C ∣ Z_{A} ∣.

Proof. Every inner-boundary vertex is incident to a crossing edge, and every crossing edge has exactly one endpoint in $A$ . The maximum degree is four. This proves the first two inequalities. Counting both endpoints of the crossing edges gives

2∣ \partial_{Λ} A ∣ \leq 4∣ Z_{A} ∣.

Substitution proves the conditional entropy statements.

Displayed formulas in this passage

unlabeled · line 318
$∣ \partial_{in} A ∣ \leq ∣ \partial_{Λ} A ∣ \leq 4∣ \partial_{in} A ∣,$
unlabeled · line 324
$∣ \partial_{Λ} A ∣ \leq 2∣ Z_{A} ∣.$
unlabeled · line 329
$S_{Ω} (A) \leq 4 C ∣ \partial_{in} A ∣, S_{Ω} (A) \leq 2 C ∣ Z_{A} ∣.$
unlabeled · line 341
$2∣ \partial_{Λ} A ∣ \leq 4∣ Z_{A} ∣.$

Lemma: Square-lattice boundary accounting

Displayed formulas in this passage

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