PreprintCC BY 4.0arXiv
On the notion of reasonable functions in Cohen--Lenstra type heuristics
Abstract
We elucidate the notion of "reasonable functions" in the context of Cohen--Lenstra heuristics. For a sequence (xi)i of independent random variables valued in an infinite discrete probability space (X,p), we investigate for what sets S of functions on X it is true that almost surely, for all f∈ S, the average of f on the sequence is equal to its expected value. We show that if p decays sufficiently quickly, as e.g. the Cohen--Lenstra distribution does, then ⋃q>1Lq(X) is such a set; while if p decays sufficiently slowly, then not even ⋂q<∞Lq(X) has this property.
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BibTeX
@misc{bartel2026notion,
title = {{On the notion of reasonable functions in Cohen--Lenstra type heuristics}},
author = {Alex Bartel and Adam Morgan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29880},
eprint = {2610.29880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29880},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the notion of reasonable functions in Cohen--Lenstra type heuristics
AU - Alex Bartel
AU - Adam Morgan
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29880
UR - https://arxiv.org/abs/2610.29880
AB - We elucidate the notion of "reasonable functions" in the context of Cohen--Lenstra heuristics. For a sequence (x_i)_i of independent random variables valued in an infinite discrete probability space (X,p), we investigate for what sets S of functions on X it is true that almost surely, for all f∈ S, the average of f on the sequence is equal to its expected value. We show that if p decays sufficiently quickly, as e.g. the Cohen--Lenstra distribution does, then ⋃_q>1L^q(X) is such a set; while if p decays sufficiently slowly, then not even ⋂_q<∞L^q(X) has this property.
ER -