Manuscript En05-B@1 · En05-B.tex · theorem · 8b626460

Theorem: Uniform Euclidean metric approximation

Uniform metric estimates on all sites · lines 610–651 · proof included or linked

En05-B@1#thm:metric, En05-B@1#eq:allmetric

For every pair of sites $x, y \in Λ_{n}$ ,

∥ z_{n} (x) - z_{n} (y) ∥_{2} \leq a_{n} d_{n} (x, y) \leq λ_{n} ∥ z_{n} (x) - z_{n} (y) ∥_{2} + B_{n} .

Consequently, for displacement at most $R$ ,

0 \leq a_{n} d_{n} (x, y) - ∥ z_{n} (x) - z_{n} (y) ∥_{2} \leq (λ_{n} - 1) R + B_{n} ⟶ 0

uniformly over all such sites.

Proof. The lower bound again follows from (eq:edgeLipschitz). Every site can be joined to one endpoint hub using at most $N_{n} + 1 = F_{n}$ edges. Choose such hubs $H, H^{'}$ for $x, y$ . Then

a_{n} d_{n} (x, H) \leq s_{n}, a_{n} d_{n} (y, H^{'}) \leq s_{n},

and the corresponding physical distances are also at most $s_{n}$ . The hub distance is therefore at most $∥ z_{n} (x) - z_{n} (y) ∥_{2} + 2 s_{n}$ . Lemma lem:hubmetric gives

a_{n} d_{n} (x, y) \leq 2 s_{n} + λ_{n} (∥ z_{n} (x) - z_{n} (y) ∥_{2} + 2 s_{n}) + s_{n} + a_{n},

which is (eq:allmetric).

Displayed formulas in this passage

En05-B@1#eq:allmetric · line 613
$∥ z_{n} (x) - z_{n} (y) ∥_{2} \leq a_{n} d_{n} (x, y) \leq λ_{n} ∥ z_{n} (x) - z_{n} (y) ∥_{2} + B_{n} .$
unlabeled · line 621
$0 \leq a_{n} d_{n} (x, y) - ∥ z_{n} (x) - z_{n} (y) ∥_{2} \leq (λ_{n} - 1) R + B_{n} ⟶ 0$
unlabeled · line 634
$a_{n} d_{n} (x, H) \leq s_{n}, a_{n} d_{n} (y, H^{'}) \leq s_{n},$
unlabeled · line 642
$a_{n} d_{n} (x, y) \leq 2 s_{n} + λ_{n} (∥ z_{n} (x) - z_{n} (y) ∥_{2} + 2 s_{n}) + s_{n} + a_{n},$

Theorem: Uniform Euclidean metric approximation

Displayed formulas in this passage

Nearby in this section