# Trickle-Down in Localization Schemes and Applications

arXiv: https://arxiv.org/abs/2407.16104 · HTML: https://mlsanity.com/paper/2407.16104/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2024.

## Built on by (most cited first, 4 of 4)

- [Diffusion Models are Secretly Exchangeable: Parallelizing DDPMs via Autospeculation](https://mlsanity.com/paper/2505.03983/) (2025)
- [Estimating Ising Models in Total Variation Distance](https://mlsanity.com/paper/2511.21008/) (2025)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)
- [On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy](https://mlsanity.com/paper/2606.12720/) (2026)

## Abstract (arXiv)

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.