  We prove that there exists a constant $\gamma_{\mathrm{crit}}\approx .17566$
such that if $G\sim \mathbb{G}(n,1/2)$ then for any $\varepsilon > 0$ with high
probability $G$ has a equipartition such that each vertex has
$(\gamma_{\mathrm{crit}}-\varepsilon)\sqrt{n}$ more neighbors in its own part
than in the other part and with high probability no such partition exists for a
separation of $(\gamma_{\mathrm{crit}}+\varepsilon)\sqrt{n}$. The proof
involves a number of tools ranging from isoperimetric results on
vertex-transitive sets of graphs coming from Boolean functions, switchings,
degree enumeration formulas, and the second moment method. Our results
substantially strengthen recent work of Ferber, Kwan, Narayanan, and the last
two authors on a conjecture of F\"uredi from 1988 and in particular prove the
existence of fully-friendly bisections in $\mathbb{G}(n,1/2)$
