Useful finite-size synchronization estimates near threshold

PDF p. 38: Section 7, last paragraph on p. 38

Assume countable choice. For a fixed $\gamma >1$, set $p_n=\gamma \log n\mathrm{/}n$ for integers $n$ for which $p_n\le 1$. In the Erdős–Rényi model, $G(n,p_n)$ has vertex set $\{1,\dots ,n\}$ and each unordered pair of distinct vertices is independently an edge with probability $p_n$. A graph is globally synchronizing when, for normalized uniform initial phase volume, its identical-oscillator Kuramoto flow in the rotating frame converges to a fully synchronized state for almost every initial state.

Find explicit, optimized upper bounds $B(n,\gamma )$ for the probability that $G(n,p_n)$ is not globally synchronizing, valid at stated finite values of $n$ and useful at practically relevant sizes throughout the prescribed logarithmic family $p_n=\gamma \log n\mathrm{/}n$ with fixed $\gamma >1$. Describe the dependence of these bounds on $\gamma$ as $\gamma \downarrow 1$. In particular, address $\gamma =2$, that is, $p_n=2\log n\mathrm{/}n$. One concrete form of the question is to give an explicit threshold $N(\gamma ,\delta )$ such that $B(n,\gamma )\le \delta$ for every integer $n\ge N(\gamma ,\delta )$ and a prescribed failure tolerance $0<\delta <1$. What counts as practically useful depends on the intended range of $n$ and tolerance; no numerical target is asserted here.

As of 2026-11-14, this item records the finite-size optimization question raised by the article, not a claimed solution or a completed review of subsequent literature. The article says that its asymptotically sharp $G(n,p)$ result was not optimized for practically relevant sizes, particularly near the connectivity threshold, and that the example $p=2\log n\mathrm{/}n$ may require an impractically large $n$.

The completed quantitative synchrony supplier gives a partial asymptotic result: under countable choice, for $0<\epsilon <1$ and $1+\epsilon <\gamma <1+{\epsilon }^{-2}$, some $C(\gamma ,\epsilon )>0$ bounds the failure probability by $C(\gamma ,\epsilon )n^{-\epsilon }$ for all sufficiently large $n$. For example, $\gamma =2$ and $\epsilon =1\mathrm{/}2$ satisfy these inequalities. This existential statement does not provide the explicit constants or finite-size threshold requested above. The supplier distinguishes its synchrony conclusion from the article's separately refuted exact-expansion probability assertion; that assertion is not used here.

The motivation and the example are attributed to Expander graphs are globally synchronizing, PDF p. 38, Section 7. The article does not establish an answer to the finite-size question in that passage.