Trickle-Down in Localization Schemes and Applications
<|reference_start|>Trickle-Down in Localization Schemes and Applications: Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems -- for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include improved mixing results for the Langevin dynamics in the $O(N)$ model, and near-linear time sampling algorithms for the antiferromagnetic and fixed-magnetization Ising models on expanders. For the latter application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials.<|reference_end|>
@article{anari2024trickle-down,
    title={Trickle-Down in Localization Schemes and Applications},
    author={Nima Anari and Frederic Koehler and Thuy-Duong Vuong},
    journal={arXiv preprint arXiv:2407.16104},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.16104},
    primaryClass={math.PR cs.DS math-ph math.MP}
}