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 <span>_i_</span>, there's a little magnetic atom with state <span>_sigma_i_</span>, which can point either up (<span>_sigma_i = +1_</span>) or down (<span>_sigma_i = -1_</span>). 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 <span>_H(sigma) = -J sum_{i text{ adjacent to } j} sigma_i sigma_j_</span> for some constant <span>_J_</span>; two adjacent atoms with the same direction contribute <span>_-J_</span> energy, while two adjacent atoms with different directions contribute <span>_+J_</span> energy.

Read: Blue -> -1 or "cold" or "down" , Red -> +1 or "hot" or "up"

With the energy function in hand, the Boltzmann distribution <span>_frac{1}{Z} e^{-H(sigma)/T} = frac{1}{Z} e^{frac{J}{T} sum_{i text{ adjacent to } j} sigma_i sigma_j}_</span> specifies the distribution of states <span>_sigma_</span> for temperature <span>_T_</span>. 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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First published:
December 12th, 2025

Source:
https://www.lesswrong.com/posts/R4qxugqxgwBsAEmzn/conditional-on-long-range-signal-ising-still-factors-locally

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Narrated by TYPE III AUDIO.

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Images from the article:

+1 or "hot" or "up"" style="max-width: 100%;" />

$Each magnetic <span>__T3A_INLINE_LATEX_PLACEHOLDER___atom_{ij}___T3A_INLINE_LATEX_END_PLACEHOLDER__</span> is represented by a node <span>__T3A_INLINE_LATEX_PLACEHOLDER___ \sigma_{ij}___T3A_INLINE_LATEX_END_PLACEHOLDER__</span>, and is adjacent to 4 immediate neighbors.$

A 50/50 mixture of two well-separated unimodal distributions.

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

LessWrong (30+ Karma)

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