Audio note: this article contains 74 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Background: The Ising Model

The Ising Model is a classic toy model of magnets. We imagine a big 2D or 3D grid, representing a crystal lattice. At each grid vertex _i_, there's a little magnetic atom with state _sigma_i_, which can point either up (_sigma_i = +1_) or down (_sigma_i = -1_). When two adjacent atoms point the same direction, their joint energy is lower than when they point different directions; atoms further apart don’t directly interact. So we write the energy function (aka Hamiltonian) as _H(sigma) = -J sum_{i text{ adjacent to } j} sigma_i sigma_j_ for some constant _J_; two adjacent atoms with the same direction contribute _-J_ energy, while two adjacent atoms with different directions contribute _+J_ energy.

With the energy function in hand, the Boltzmann distribution _frac{1}{Z} e^{-H(sigma)/T} = frac{1}{Z} e^{frac{J}{T} sum_{i text{ adjacent to } j} sigma_i sigma_j}_ specifies the distribution of states _sigma_ for temperature _T_. Crucially [...]

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Outline:

(00:23) Background: The Ising Model

(03:00) Hot Claim: Conditional On Long-Range Signal At Low Temperature, Ising Approximately Factors Over The Same Graph

(03:41) Proof Sketch

(03:45) First Piece: Using Maxent To Pick Out A Mode (In General)

(08:13) Second Piece: Using Maxent To Pick Out A Mode (Ising at Low Temp)

(09:57) Third Piece: Approximate Factorization Yay!

(10:25) Why Is This Interesting?

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