I just don’t understand how it would be possible over any time period, doesn’t matter if it’s a billion years or even 10^10^10^10… years. There are conservation laws governing how particles can pop into existence and them all popping into existence in just the right place is only even remotely plausible if you ignore the antiparticles that are formed in the same space.
Forget about the particles for a second. This is about fundamental quantum fields. The fields can randomly fluctuate from the vacuum state into a more ordered state where virtual particles are created.
The conjecture here is that a given volume of space must, at some point, randomly evolve into the quantum state you're interested. When the quantum fields align into the same state that a bunch of particles would represent, those particles appear out of the vacuum.
The trick that makes this work is that conservation laws don't apply on very small time scales. That's how virtual particles work after all. The energy can only be temporarily borrowed from the vacuum, unless you pay the energy cost to make that particle 'real' by destroying its virtual pair (see Hawking radiation).
You might imagine TV static, just random visual noise. There's no real reason the randomness can't line up to produce one single coherent frame before decohering. Just imagine that in 3 (or 11) dimensions.
I don't think an antiparticle pair is a strict requirement for virtual particles either. As long as energy is conserved on macro timescales, the universe doesn't really care what state the quantum fields are in.
Conservation of energy is only considered true "locally", but not globally.
There are many examples of this in nature, believe it or not.
If you wanted to have a system that perfectly conserved energy, how would you expect to measure its existence, or changes in state, anyway?
Besides, even if you assume conservation of particle number, it only has to be eventual conservation.
Suppose I borrow some energy from "over here" temporarily and accidentally assemble them into a perfect brain for a second, then they can disperse again, if they like.
A bigger question is, if energy were conserved globally, where exactly did those handy particles come from in the first place anyway?
Among the de Sitter invariant states for massive fields is the Euclidean or Bunch-Davies vacuum, which allows for pair production with separations greater than the horizon distance. Since our universe seems to be marching towards a state highly similar to the Euclidean vacuum, such pairs seem (a) physically plausible (b) and with some low probability could arise in groups which (c) because they are interacting particles, might nucleate.
Nucleated objects could be long-lasting, which blunts the Boltzmann brain (Bb) picture, as under fluctuation theory the Bb is ephemeral and time-reversible. On the other hand, it's historically been attractive to think of the nucleation of a inflating patch of spacetime (with low enough entropy that structure like galaxies might form as it expands and fragments gravitationally).
