Noise Stability and Central Limit Theorem for Random Effective Resistance

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Model assumes IID resistances in [1,Λ] on Z^d and identifies effective conductance ~n^{d-2} (so resistance ~n^{2-d}). Variance has expected lower bound ~n^{4-3d}; analysis shows fluctuations come from low-order Walsh components and gives quantitative control via the Walsh decomposition, reciprocity law for second derivatives of the minimal current, and Lindeberg/Chatterjee second-order bounds. A Gaussian approximation holds on the torus and a full CLT proven for d=1, with failure modes tied to boundary effects and lack of current uniqueness.

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1. 0:00Start
2. 3:26Effective resistance and effective conductance
3. 10:47Definitions
4. 15:15Currents and effective resistance
5. 17:40Strategy
6. 17:51Walsh decomposition
7. 27:25Proofs of (1) and (2)
8. 34:14Proof of (2)
9. 37:11Consequence of (2): Gaussian approximation
10. 41:12Adaptation of the preceding setting?
11. 50:32Homogeneity on the torus