Negative Dependence and Lorentzian Polynomials

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" summary

Claims that the random cluster / q‑state Potts partition function is Lorentzian for q ≤ 1, proves the support is a matroid/poly‑matroid and reduces many checks to rank‑2 cases via conditioning (partial derivatives). Tools are multivariate partition functions, M‑convexity, and stability/Lorentzian theory. Consequences: the rank generating sequence satisfies Newton’s inequalities and the model has almost negative correlations (correlation bounds hold up to a factor 2), but full negative association remains unproven.

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" chapters (7)

1. 0:00Introduction to polynomials
2. 3:08Negative dependence properties
3. 9:47Random cluster model
4. 18:09Lorentzian polynomials
5. 26:58Matroid and M-convex sets
6. 36:56Lorentzian characterization
7. 44:02Applications and results

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