Theorem F — Spectral Stability

Prerequisites: Theorem A (energy isolation), Chapter 8.

Theorem F. Let $\|\delta W\|_\infty:=\max_{i,j}|W'_t(i,j)-W_t(i,j)|$.

(i) $|\phi'_t(F)-\phi_t(F)|\le C_1\|\delta W\|_\infty|V|^2/\mathrm{vol}_t(F)$, where $C_1=2(1+\theta)\le 4$.

(ii) If $\phi_t(F)\le\theta-\epsilon$ with $\epsilon>0$, then $F\in\mathfrak{F}'_t$ provided $\|\delta W\|_\infty<\epsilon\cdot\mathrm{vol}_t(F)/(C_1|V|^2)$.

Proof. (Complete; reproduced from Chapter 8.)

(i) Write $\phi=\mathrm{cut}/\mathrm{vol}$ (under (F1), denominator is $\mathrm{vol}(F)$).

Numerator perturbation: $|\Delta_{\mathrm{cut}}|\le|F|\cdot|\bar F|\cdot\|\delta W\|_\infty\le|V|^2\|\delta W\|_\infty$.

Denominator perturbation: $|\Delta_{\mathrm{vol}}|\le|F|\cdot|V|\cdot\|\delta W\|_\infty\le|V|^2\|\delta W\|_\infty$.

Numerator: $|\Delta_c\cdot\mathrm{vol}-\mathrm{cut}\cdot\Delta_v|\le\|\delta W\|_\infty|V|^2(\mathrm{vol}+\mathrm{cut})\le\|\delta W\|_\infty|V|^2(1+\theta)\mathrm{vol}$.

Denominator: $(\mathrm{vol}+\Delta_v)\cdot\mathrm{vol}\ge\mathrm{vol}^2/2$ for $|\Delta_v|\le\mathrm{vol}/2$.

Result: $|\phi'-\phi|\le 2(1+\theta)\|\delta W\|_\infty|V|^2/\mathrm{vol}=C_1\|\delta W\|_\infty|V|^2/\mathrm{vol}$.

(ii) Set $\delta_0=\epsilon/(C_1|V|^2/\mathrm{vol}_t(F))$. Then $|\phi'-\phi|<\epsilon$, so $\phi'_t(F)<\theta$. Volume condition preserved similarly. $\square$

Explicit constant. $C_1=2(1+\theta)$. For $\theta=0.1$, $C_1=2.2$. The perturbation size requirement is $\|\delta W\|_\infty<\frac{\epsilon\,\mathrm{vol}(F)}{2.2\,|V|^2}$, or equivalently $\|\delta W\|_\infty\cdot\frac{|V|^2}{\mathrm{vol}(F)}<\frac{\epsilon}{2.2}$.