  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.
