Topic: stochastic localization

Sampling uniformly from a convex body links randomized volume computation to the Kannan–Lovász–Simonovits conjecture. This talk identifies geometric quantities controlling sampler mixing and introduces stochastic localization as a way to bound them, including a simple argument yielding a polylogarithmic, though nonoptimal, bound.

Stochastic localization has become a tool for high-dimensional geometry, sampling and diffusion-model analysis. Motivated by mirror descent and non-Euclidean sampling, this talk generalizes the process by replacing Gaussian regularization with multiples of log-Laplace transforms. It introduces the construction, its analysis, applications and remaining questions.

A polynomial-time algorithm samples the Sherrington–Kirkpatrick Gibbs distribution with negligible total-variation error for inverse temperatures below one half. The talk combines stochastic localization, Jarzynski's equality and rejection sampling with covariance and functional-inequality estimates. It also derives a weak Poincaré inequality that simplifies the construction and relates the guarantees to earlier Wasserstein results.