2112.12851
ON THE LIMITING DISTRIBUTION OF FREE PATH LENGTHS FOR FLAT SURFACES WITH CIRCULAR OBSTACLES
Diaaeldin Taha
incompletemedium confidence
- Category
- math.DS
- Journal tier
- Note/Short/Other
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s argument has the right overall structure (approximation by segment obstacles; renormalization; circle-average equidistribution) and definitions match. However, its proof of the equidistribution step cites Eskin–Masur (2001) in a way that appears insufficient for the general SL(2,R)-ergodic measure μ asserted, and it does not address the discontinuity of the test function needed for weak-* convergence. The model’s solution supplies the appropriate modern equidistribution input (Eskin–Mirzakhani–Mohammadi) and a μ-a.e. continuity argument, and it gives a sharper geometric comparison inequality, thereby closing the gaps.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} note/short/other
\textbf{Justification:}
This short note elegantly synthesizes known connections between free path length distributions and zippered-rectangle heights, extending them to general flat surfaces. The core ideas are correct, and the argument structure is clean. However, the equidistribution input is under-cited (a reference to Eskin–Masur 2001 appears insufficient for general affine invariant measures), and the needed μ-a.e. continuity for the test function is not addressed. These can be fixed with brief additions.