Condensation Transition in Entropy-Constrained Probability Spaces

Abstract

Probability spaces are known to undergo a concentration of measure in high dimensions. In particular, if a $K$-dimensional probability distribution is sampled from a $(K-1)$-dimensional simplex with uniform measure, with high probability its empirical distribution of components is logarithmic. Here, we extend this result to the subset of distributions populating level sets of de simplex defined by a fixed Shannon entropy $H_0$. We show that the empirical distribution of components of the large majority of sampled distributions undergoes a phase transition. A critical entropy $H_c \approx \log K - 1 + \gamma $ exists, below which the overwhelming majority of sampled distributions are found in a condensed state, in which a single component captures a macroscopic fraction of the total probability, while the remaining components form a homogeneous incompressible fluid background. We derive the magnitude of the critical entropy, as well as the empirical distribution of components both above and below threshold. These results provide insight into how the level sets of fixed entropy in the simplex are populated.