Entropy, Ultralimits and the Poisson boundary

Elad Sayag, Yehuda Shalom

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that the RN-factor of the U-Abel ultralimit of (G, µa) gives a topological model of the Poisson–Furstenberg boundary via a C*-algebraic universality/extension argument, culminating in Theorem 5.9. The candidate’s solution establishes the same identification but through a different route: it uses uniform BQI bounds, KL data-processing, and an Abel-limit/likelihood-ratio analysis on path space to show equality of the KL profiles for all g and then invokes RN minimality. Both arguments are coherent and align on the main result; the model’s proof is more information-theoretic, while the paper’s is via ultralimit functoriality and measure-preserving extensions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript develops a clean and flexible ultralimit framework for BQI G-spaces and uses it to construct the Poisson–Furstenberg boundary as an RN-factor of an ultralimit of actions on the group with Abel measures. The approach is natural and seems robust for further applications (for instance, to quantitative entropy questions). The proof of the boundary identification is concise and elegant, relying on a universality property and standard facts about measure-preserving extensions of the boundary. Some details (e.g., explicit control or examples of uniform BQI bounds, and a brief discussion of the continuity lemma’s scope) could be amplified, but overall the paper is clearly written and correct.