Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G \times K_2$, which has vertex set $V\times \{0,1\}$ with “horizontal” edges inherited from $G$ and additional “vertical” edges connecting $(w,0)$ and $(w,1)$ for each $w \in V$. Kasteleyn’s Bunkbed conjecture states that for each $u,v \in V$ and $p\in [0,1]$, the vertex $(u,0)$ is at least as likely to be connected to $(v,0)$ as to $(v,1)$ under Bernoulli-$p$ bond percolation on the bunkbed graph. We prove that the conjecture holds in the $p \uparrow 1$ limit in the sense that for each finite graph $G$ there exists $\varepsilon (G)\gt 0$ such that the bunkbed conjecture holds for $p \geqslant 1-\varepsilon (G)$.

The origin of ionic conductivity in bulk lithium lanthanum titanate, a promising solid electrolyte for Li-ion batteries, has long been under debate, with experiments showing lower conductivity than predictions. Using first-principles-based calculations, we find that experimentally observed type I boundaries are more stable compared with the type II grain boundaries, consistent with their observed relative abundance. Grain boundary stability appears to strongly anti-correlate with the field strength as well as the spatial extent of the space charge region. Ion migration is faster along type II grain boundaries than across, consistent with recent experiments of increased conductivity when type II densities were increased.