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ABSTRACT A key set‐theoretic “spread” lemma has been central to two recent celebrated results in combinatorics: the recent improvements on the sunflower conjecture by Alweiss, Lovett, Wu, and Zhang; and the proof of the fractional Kahn–Kalai conjecture by Frankston, Kahn, Narayanan, and Park. In this work, we present a new proof of the spread lemma, that—perhaps surprisingly—takes advantage of an explicit recasting of the proof in the language of Bayesian inference. We show that from this viewpoint the reasoning proceeds in a straightforward and principled probabilistic manner, leading to a truncated second moment calculation which concludes the proof. 

Abstract Entropic Brenier maps are regularized analogues of Brenier maps (optimal transport maps) which converge to Brenier maps as the regularization parameter shrinks. In this work, we prove quantitative stability bounds between entropic Brenier maps under variations of the target measure. In particular, when all measures have bounded support, we establish the optimal Lipschitz constant for the mapping from probability measures to entropic Brenier maps. This provides an exponential improvement to a result of Carlier, Chizat, and Laborde (2024). As an application, we prove near-optimal bounds for the stability of semi-discrete unregularized Brenier maps for a family of discrete target measures. 

ABSTRACT The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high‐dimensional settings. However, existing analyses of the algorithm for strongly log‐concave distributions suggest that, as the dimension of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the error for the full set of variables, the behavior for asmall numberof variables can be significantly better: A number of iterations proportional to , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all ‐marginals. We refer to this effect asdelocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log‐concave distributions with certain sparse interactions. Our analysis relies on a novel metric to measure convergence. A key technical challenge we address is the lack of a one‐step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.