Exponential decay of intersection volume with applications on list--decodability and Gilbert--Varshamov type bound

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming, Johnson spaces and symmetric groups decay exponentially as their centers drift apart. To verify condition (iii), we prove some large deviation inequalities `on a slice' for functions with Lipschitz conditions. We then use these estimates on intersection volumes to $\bullet$ obtain a sharp lower bound on list-decodability of random $q$-ary codes, confirming a conjecture of Li and Wootters; and $\bullet$ improve the classical bound of Levenshtein from 1971 on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy. Our probabilistic point of view also offers a unified framework to obtain improvements on other Gilbert--Varshamov type bounds, giving conceptually simple and calculation-free proofs for $q$-ary codes, permutation codes, and spherical codes. Another consequence is a counting result on the number of codes, showing ampleness of large codes.