]{}, and then summing over all these combinations. For example, the contribution from the case in which ${\vbv_2-y{|\!|\!|}}^q$ and ${\vbv'_1-y{|\!|\!|}}^{2q}$ in [(\[eq:Pj-bd\])]{} are replaced, respectively, by ${\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q$ and ${\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1-v{|\!|\!|}}^q$, is bounded by $$\begin{aligned}
&\sum_{v_2,v'_1}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1
 -v{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q{\vbv'_1-v_2{|\!|\!|}}^q}\,\frac{O(\theta_0^2)^{j
 -1}}{{\vbx-v'_1{|\!|\!|}}^q{\vbx-v_2{|\!|\!|}}^{2q}}{\nonumber}\\
&\leq\sum_{v'_1}\frac{O(\theta_0^2)^j}{{\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1
 -v{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbx-v'_1{|\!|\!|}}^{2q}}{\nonumber}\\
&\leq\frac{O(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q
 {\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.\end{aligned}$$ The other $(2j-1)j-1$ contributions can be estimated similarly, with the same form of the bound. This completes the proof of [(\[eq:P”j-bd\])]{} and thus Proposition \[prp:GimpliesPix\].

Bounds for finite-range models {#ss:proof-nn}
------------------------------

First, we prove [(\[eq:pi-sumbd\])]{} and Proposition \[prp:exp-bootstrap\](iii) assuming [(\[eq:IR-kbd\])]{}. Then, we prove [(\[eq:pi-kbd\])]{} assuming [(\[eq:IR-kbd\])]{} and [(\[eq:IR-xbdNN\])]{} to complete the proof of Propositions \[prp:GimpliesPik\].

By applying [(\[eq:G-delta-bd\])]{} to the bound [(\[eq:piNbd\])]{} on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$, it is easy to show that, for $r=0,2$, $$\begin{aligned}
{\label{eq:pi0-rthmombd}}
\sum_x|x|^r\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq\delta_{r,0}+\sum_{x\ne o}
 |x|^r{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3&\leq\delta_{r,0}+\Big(
 \sup_{x\ne o}|x|^rG(x)\Big)\sum_{x\ne o}(\tau D*G)(x)\,G(x){\nonumber}\\
&\leq\delta_{r,0}+(d\sigma^2)^{\delta_{r,2}}O(\theta_0)^2.\end{aligned}$$

For $i\ge1$, by using the diagrammatic bound [(\[eq:piNbd\])]{} and