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McShane's identity and the Brownian loop measure
Abstract
We combine McShane's identity for a once-punctured torus with the length-spectrum identity of Wang and Xue. An exact identity in each primitive homology class yields a finite limit for the difference of two divergent length-spectrum sums over self-intersecting hyperbolic geodesics. The limit is 1/8 and is equivalent to McShane's identity. We also derive higher-iteration identities governed by integer divisors.
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BibTeX
@misc{hedenmalm2026mcshanes,
title = {{McShane's identity and the Brownian loop measure}},
author = {Haakan Hedenmalm},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08687},
eprint = {2611.08687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08687},
}
RIS
TY - UNPB
TI - McShane's identity and the Brownian loop measure
AU - Haakan Hedenmalm
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08687
UR - https://arxiv.org/abs/2611.08687
AB - We combine McShane's identity for a once-punctured torus with the length-spectrum identity of Wang and Xue. An exact identity in each primitive homology class yields a finite limit for the difference of two divergent length-spectrum sums over self-intersecting hyperbolic geodesics. The limit is 1/8 and is equivalent to McShane's identity. We also derive higher-iteration identities governed by integer divisors.
ER -