Algorithmic Methods in Statistical Physics



This reading group will focus on understanding the technical details related to proving the Parisi variational principle, and its role in modern algorithmic applications of spin glass theory.

Logistics

Location John Crerar Library (JCL) 354

Time Every Tuesday at 10:30am - 12pm

Logistics Please check this webpage periodically for updates and material

Topics Outline

  • Introduction to the replica method, and its use in deriving Parisi’s ansatz for the Sherrington-Kirkpatrick model.

  • Demonstrate Panchenko’s lowerbound argument for the Parisi variational principle. Focus specifically on the geometric intuition for the Gibbs distribution. Showcase fundamental tools for the argument:

    • Gaussian integration by parts, Gaussian concentration, and Stein’s Lemma (the Gaussian toolbox)
    • Exchangeability via Gram-de Finetti arrays (Aldous-Hoover, Dovbysh-Sudakov)
    • Poisson processes and the Ruelle Probability Cascades
    • Pure state decompositions

  • [Optional] Showcase the Guerra-Toninelli (smart-path) interpolation technique, and demonstrate their upperbound argument for the Parisi variational principle.

  • Discuss the Gibbs distribution for spherical p-spin: pure states, thin bands, and their uses in constructing Subag’s algorithm for finding the ground state.

  • Present Montanari’s algorithm for the Sherrington-Kirkpatrick model and where it takes inspiration from Subag’s algorithm.

Reading List

Schedule