“Conditional On Long-Range Signal, Ising Still Factors Locally” by johnswentworth, David Lorell

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Recorded14 Dec 2025

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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.Read: Blue -> -1 or "cold" or "down" , Red -> +1 or "hot" or "up" 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 [...] ---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? --- First published: December 12th, 2025 Source: https://www.lesswrong.com/posts/R4qxugqxgwBsAEmzn/conditional-on-long-range-signal-ising-still-factors-locally --- Narrated by TYPE III AUDIO. ---Images from the article: -1 or "cold" or "down" , Red - +1 or "hot" or "up"" style="max-width: 100%;" /__T3A_INLINE_LATEX_PLACEHOLDER___atom_{ij}___T3A_INLINE_LATEX_END_PLACEHOLDER__ is represented by a node __T3A_INLINE_LATEX_PLACEHOLDER___ \sigma_{ij}___T3A_INLINE_LATEX_END_PLACEHOLDER__, and is adjacent to 4 immediate neighbors." style="max-width: 100%;" /Apple Podcasts and Spotify do not show images in the episode description. Try Pocket Casts, or another podcast app.

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