A positive minimum degree envelope at every fixed supercritical ratio

Write a degree as $S={\sum }_{i=1}^{n-1}{X}_i$ using its incident edge indicators. For fixed real $t\mathrm{\ne }0$, the map $x\mapsto e^{tx}$ is injective on $\{0,1\}$ by [F2]; every joint atom for any subfamily of $(e^{t{X}_i}{)}_i$ is therefore the corresponding joint atom for $(X_i{)}_i$. Its probability factors by [F3], proving mutual independence of the transformed variables. For $t=0$ they are all constant $1$, so their joint atom probabilities also factor. All factors are finite-valued on the finite graph probability space, so their expectations are finite. Product factorization in [F1] and the exponential addition formula in [F2] now give $\mathbb{E}{e}^{tS}=(1+p(e^t-1){)}^{n-1}\le \exp (\mu (e^t-1))$. The elementary bound $1+x\le e^x$ follows from the derivative of $e^x-1-x$, whose minimum is at zero. For $0<r<1$, apply Markov to $r^S$ on $S\le r\mu$ and obtain $\mathbb{P}(S\le r\mu )\le \exp (-\mu h(r)),h(r)=1-r+r\log r\mathrm{.}$ Indeed $\mathbb{E}{r}^S\le e^{\mu (r-1)}$ and the threshold is $r^{r\mu }$.

As $r\downarrow 0$, $r\log r\to 0$ by substituting $t=1\mathrm{/}r$ in [F2], so $h(r)\to 1$. Choose $r\in (0,1)$ with ${\gamma }_{0}h(r)>1$ and set $a=r\mathrm{/}2$, $\eta =({\gamma }_{0}h(r)-1)\mathrm{/}2>0$. For $n\ge 2$, $ad\le r\mu$; eventually $(1-1\mathrm{/}n){\gamma }_{0}h(r)\ge 1+\eta$. A union bound then gives $\mathbb{P}(\mathrm{\exists }v:S_v<ad)\le n{e}^{-\mu h(r)}\le n^{-\eta }\mathrm{.}$ The cutoff depends on ${\gamma }_0,r$ only, since $\mu \ge (1-1\mathrm{/}n){\gamma }_0\log n$.

Take $t=\log 2>0$, $b=4\mathrm{/}\log 2$. The same moment bound gives $\mathbb{P}(S\ge bd)\le e^{\mu -bd\log 2}\le e^{-3d}$. The union probability for $S_v>bd$ is therefore at most $n{e}^{-3d}\le n^{1-3{\gamma }_0}\le n^{-2}$. Positivity of $b-1$ follows from $0<\log 2<1$, for example by integrating $1\mathrm{/}x$ from 1 to 2. Add the two exceptional probabilities. No independence between different degrees is used.