Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

=== Assistant: Here is a single LaTeX snippet drop-in that packages the full “solve for ϕC\phiCϕC” calculation, plus the two regimes a… ===
Here is a single LaTeX snippet (drop-in) that packages the full “solve for ϕ(C)\phi(C)ϕ(C)” calculation, plus the two regimes and the ϕ(C)=O(ρ)\phi(C)=O(\rho)ϕ(C)=O(ρ) headline under typical local-cluster scalings. It’s written to fit directly after your definition of Δ(C)\Delta(C)Δ(C) (or anywhere in that section). open-problem-fountoulakis22a

<syntaxhighlight lang="latex">% ------------------------------------------------------------
% Solving the Delta(C)-condition for conductance, with regimes.
% ------------------------------------------------------------
\subsubsection*{How small must $\phi(C)$ be? Solving the $\Delta(C)$ condition}

Recall the sufficient slack condition used in the outside-margin bound:
\begin{equation}\label{eq:Delta_slack_condition}
\Delta(C)\le (1-\eta)\rho\alpha\sqrt{d_{\mathrm{out}}},
\qquad \eta\in(0,1),
\end{equation}
where
\[
\Delta(C)
=
\frac{1-\alpha}{2\mu_C}\,
\sqrt{\frac{\phi(C)\mathrm{vol}(C)}{d_{\mathrm{out}}\,d_{\mathrm{in}}}}\,
\left(\frac{\alpha}{\sqrt{d_v}}+\rho\alpha\sqrt{\mathrm{vol}(C)}\right).
\]

\begin{lemma}[Conductance threshold implied by \eqref{eq:Delta_slack_condition}]
\label{lem:phi_threshold_exact}
The condition \eqref{eq:Delta_slack_condition} is equivalent to
\begin{equation}\label{eq:phi_max_exact}
\phi(C)\;\le\;\phi_{\max}(C)
:=\frac{4\mu_C^2(1-\eta)^2\rho^2\,d_{\mathrm{out}}^2\,d_{\mathrm{in}}}
{(1-\alpha)^2\,\mathrm{vol}(C)\,\Bigl(\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\Bigr)^2}.
\end{equation}
\end{lemma}

\begin{proof}
Start from \eqref{eq:Delta_slack_condition} and substitute the definition of $\Delta(C)$.
Cancel the positive factor $\alpha$ on both sides to get
\[
\frac{1-\alpha}{2\mu_C}\,
\sqrt{\frac{\phi(C)\mathrm{vol}(C)}{d_{\mathrm{out}}\,d_{\mathrm{in}}}}\,
\left(\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\right)
\le
(1-\eta)\rho\sqrt{d_{\mathrm{out}}}.
\]
Rearrange to isolate the square root:
\[
\sqrt{\frac{\phi(C)\mathrm{vol}(C)}{d_{\mathrm{out}}\,d_{\mathrm{in}}}}
\le
\frac{2\mu_C}{1-\alpha}\cdot
\frac{(1-\eta)\rho\sqrt{d_{\mathrm{out}}}}{\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}}.
\]
Squaring both sides and solving for $\phi(C)$ gives \eqref{eq:phi_max_exact}.
\end{proof}

\begin{remark}[Two useful regimes]\label{rem:phi_regimes}
Let $a:=\frac{1}{\sqrt{d_v}}$ and $b:=\rho\sqrt{\mathrm{vol}(C)}$, so the denominator in
\eqref{eq:phi_max_exact} is $(a+b)^2$.
Two common regimes give simpler sufficient (but slightly stronger) conditions:
\begin{enumerate}
\item \textbf{Seed-dominated regime ($b\le a$).}
If $\rho\sqrt{\mathrm{vol}(C)}\le 1/\sqrt{d_v}$, then $(a+b)^2\le (2a)^2=4/d_v$.
Thus a sufficient condition for \eqref{eq:Delta_slack_condition} is
\begin{equation}\label{eq:phi_seed_sufficient}
\phi(C)\;\le\;
\frac{\mu_C^2(1-\eta)^2\rho^2\,d_{\mathrm{out}}^2\,d_{\mathrm{in}}\,d_v}
{(1-\alpha)^2\,\mathrm{vol}(C)}.
\end{equation}

\item \textbf{Penalty-dominated regime ($a\le b$).}
If $1/\sqrt{d_v}\le \rho\sqrt{\mathrm{vol}(C)}$, then $(a+b)^2\le (2b)^2=4\rho^2\mathrm{vol}(C)$.
Thus a sufficient condition for \eqref{eq:Delta_slack_condition} is
\begin{equation}\label{eq:phi_penalty_sufficient}
\phi(C)\;\le\;
\frac{\mu_C^2(1-\eta)^2\,d_{\mathrm{out}}^2\,d_{\mathrm{in}}}
{(1-\alpha)^2\,\mathrm{vol}(C)^2}.
\end{equation}
\end{enumerate}
\end{remark}

\begin{remark}[Why $\phi(C)=O(\rho)$ in the typical local-cluster scaling]\label{rem:phi_O_rho}
A common locality regime assumes
\[
\mathrm{vol}(C)\le \frac{V_0}{\rho},
\qquad
d_{\mathrm{in}}\ge d_0,
\qquad
d_v\ge d_0,
\qquad
\mu_C\ge \mu_0,
\qquad
d_{\mathrm{out}}\ge \frac{c_0}{\alpha^2\rho},
\]
for constants $V_0,d_0,\mu_0,c_0>0$.
For small $\rho$, one typically has $\rho\sqrt{\mathrm{vol}(C)}\le \rho\sqrt{V_0/\rho}=\sqrt{V_0\rho}\le 1/\sqrt{d_0}$,
so the seed-dominated regime in Remark~\ref{rem:phi_regimes} applies.
Then \eqref{eq:phi_seed_sufficient} yields the explicit scaling
\[
\phi(C)\;\le\;
\frac{\mu_0^2(1-\eta)^2\,c_0^2\,d_0^2}{\alpha^4(1-\alpha)^2\,V_0}\;\rho,
\]
which is of the form $\phi(C)\le c\,\rho$ with a constant $c$ depending only on
$(\mu_0,d_0,c_0,V_0,\alpha,\eta)$.
\end{remark}
% ------------------------------------------------------------

</syntaxhighlight>

If you want, I can also add a one-line corollary right after this snippet that explicitly states: “If ϕ(C)≤ϕmax⁡(C)\phi(C)\le\phi_{\max}(C)ϕ(C)≤ϕmax(C) then Δ(C)≤(1−η)ραdout\Delta(C)\le(1-\eta)\rho\alpha\sqrt{d_{\mathrm{out}}}Δ(C)≤(1−η)ραdout and hence γCˉ≥ηραdout\gamma_{\bar C}\ge \eta\rho\alpha\sqrt{d_{\mathrm{out}}}γCˉ≥ηραdout” (linking directly back to your theorem).