Improved rapid phase ordering for Ising dynamics on random regular graphs

We study Glauber dynamics for the Ising model on random $d$-regular graphs with $n$ vertices at low temperature, in the regime $\beta \ge C\mathrm{/}d$ where $C,d$ are sufficiently large constants. In this regime, the full mixing time is exponential in $n$, because of the bottleneck between the two dominant phases: configurations with a majority of plus spins and configurations with a majority of minus spins. We show that, for sufficiently large $d$, if the dynamics is initialized with an excess of at least ${\epsilon }_{C,d}n$ plus spins over minus spins, then it quasi-equilibrates in time $O(\log n)$ to the Gibbs measure conditioned on positive magnetization. In addition, the required bias ${\epsilon }_{C,d}$ can be chosen so that ${\epsilon }_{C,d}\to 0$ as $d,C\to \mathrm{\infty }$. Previous bound of \cite{GSS25+} in the Ising and large-$d$ setting required $\beta \gtrsim (\log d)\mathrm{/}d$. A key difficulty is that, even from the all-plus initialization, one must control the propagation of negative information in spacetime in a low-temperature model with strong local correlations. We address this by developing a new bootstrap approach to proving subcriticality of disagreement clusters under the grand coupling.