My motivation for studying these systems was to discuss a toy model similar to spin glasses, but without having to introduce the Hamiltonian, so non-p

Probabilistic spin glass - Conclusion

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2024-10-19 11:00:04

My motivation for studying these systems was to discuss a toy model similar to spin glasses, but without having to introduce the Hamiltonian, so non-physicist readers can also follow along. I'm not actually sure how well this "probabilistic spin glass" resembles spin glasses studied by physicists. (I stopped being a real physicist 10+ years ago.)

The idea was to take a $N \times N$ sized spin system, ie. each cell has two states, ↑ and ↓, or 0 and 1. If each spin is indepdendent of its neighbours, then the whole system is just a list of indepdendent random variables, the geometry doesn't matter. To make it interesting, I made the spins dependent on the neighbours: for spin $s$ and neighbour $n$: $P(s=↑ | n=↑) = P(s=↓ | n=↓) = p$. So the spins align with probability $p$ and are opposite with probability $1-p$.

However, later I noticed that the construction method is not symmetric. Ie. there are grids that are "the same" (grid A and B are the same if you can go from one to the other by some combination of flipping all bits at once, flipping the grid up-down or left-right, or rotating the grid in either direction).