Proposition 4.17 (00S4) · ScalingStacks

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Proposition 4.17. (Local potential upper bound)

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Inside $U^{s,*}_{w}\subset X_{s}$ , for $\langle m,w\rangle=1$ , the local potentials satisfy $\varphi_{m}-\psi_{m}\leq Cs^{-1/2},$ or equivalently $\varphi-\psi\leq Cs^{-1/2}$ .
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Inside $U^{s,face}_{w}$ , the local potentials satisfies $\varphi_{0}-\psi_{0}\leq Cs^{-1/2}$ , or equivalently $\varphi-\psi\leq Cs^{-1/2}$ .

Proof. In the star type region case, by Cor. 4.7, we have the upper bound $\varphi_{m}-\bar{\phi}_{m,w}\leq Cs^{-1/2}$ . By Prop. 4.14 and Cor. 4.15, in $U^{s,*}_{w}$ we can replace $\bar{\phi}_{m,w}$ by $\psi_{m}$ up to an error bounded by $Cs^{-1/2}$ , hence the claim. The face type region follows the same argument, without the translational invariance statement of Cor. 4.15. ∎