Bautista Arenaza | Searcharxiv

arXiv2026

Condensation Transition in Entropy-Constrained Probability Spaces

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 $H_c \approx \log K - 1 + γ$ 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.

cond-mat.stat-mech↗

arXiv2024

Expansion of net correlations in terms of partial correlations

The marginal correlation between two variables is a measure of their linear dependence. The two original variables need not interact directly, because marginal correlation may arise from the mediation of other variables in the system. The underlying network of direct interactions can be captured by a weighted graphical model. The connection between two variables can be weighted by their partial correlation, defined as the residual correlation left after accounting for the linear effects of mediating variables. While matrix inversion can be used to obtain marginal correlations from partial correlations, in large systems this approach does not reveal how the former emerge from the latter. Here we present an expansion of marginal correlations in terms of partial correlations, which shows that the effect of mediating variables can be quantified by the weight of the paths in the graphical model that connect the original pair of variables. The expansion is proved to converge for arbitrary probability distributions. The graphical interpretation reveals a close connection between the topology of the graph and the marginal correlations. Moreover, the expansion shows how marginal correlations change when some variables are severed from the graph, and how partial correlations change when some variables are marginalised out from the description. It also establishes the minimum number of latent variables required to replicate the exact effect of a collection of variables that are marginalised out, ensuring that the partial and marginal correlations of the remaining variables remain unchanged. Notably, the number of latent variables may be significantly smaller than the number of variables that they effectively replicate. Finally, for Gaussian variables, marginal correlations are shown to be related to the efficacy with which information propagates along the paths in the graph.

stat.ME↗