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Graphene is increasingly explored as a possible platform for developing novel separation technologies^{,,,,,,,,,,,,,,,,,,}. This interest has arisen because it is a maximally thin membrane that, once perforated with atomic accuracy, may allow ultrafast and highly selective sieving of gases, liquids, dissolved ions and other species of interest^,,,,,,,,,,,. However, a perfect graphene monolayer is impermeable to all atoms and molecules under ambient conditions^,,,,,,: even hydrogen, the smallest of atoms, is expected to take billions of years to penetrate graphene’s dense electronic cloud^,,,. Only accelerated atoms possess the kinetic energy required to do this^,. The same behaviour might reasonably be expected in the case of other atomically thin crystals^,. Here we report transport and mass spectroscopy measurements which establish that monolayers of graphene and hexagonal boron nitride (hBN) are highly permeable to thermal …