Geometry of random Cayley graphs of Abelian groups
Bibliographic record
Abstract
Consider the random Cayley graph of a finite Abelian group G with respect to k generators chosen uniformly at random, with 1≪logk≪log|G|. Draw a vertex U∼Unif(G). We show that the graph distance dist(id,U) from the identity to U concentrates at a particular value M, which is the minimal radius of a ball in Zk of cardinality at least |G|, under mild conditions. In other words, the distance from the identity for all but o(|G|) of the elements of G lies in the interval [M−o(M),M+o(M)]. In the regime k≳log|G|, we show that the diameter of the graph is also asymptotically M. In the spirit of a conjecture of Aldous and Diaconis (Technical Report 231 (1985)), this M depends only on k and |G|, not on the algebraic structure of G. Write d(G) for the minimal size of a generating subset of G. We prove that the order of the spectral gap is |G|−2/k when k−d(G)≍k and |G| lies in a density-1 subset of N or when k−2d(G)≍k. This extends, for Abelian groups, a celebrated result of Alon and Roichman (Random Structures Algorithms 5 (1994) 271–284). The aforementioned results all hold with high probability over the random Cayley graph.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".