Cutoff for Almost All Random Walks on Abelian Groups
Bibliographic record
Abstract
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$; denote it $G_k$. A conjecture of Aldous and Diaconis (1985) asserts, for $k \gg \log |G|$, that the random walk on this graph exhibits cutoff. Further, the cutoff time should be a function only of $k$ and $|G|$, to sub-leading order. This was verified for all Abelian groups in the '90s. We extend the conjecture to $1 \ll k \lesssim \log |G|$. We establish cutoff for all Abelian groups under the condition $k - d(G) \gg 1$, where $d(G)$ is the minimal size of a generating subset of $G$, which is almost optimal. The cutoff time is described (abstractly) in terms of the entropy of random walk on $\mathbb Z^k$. This abstract definition allows us to deduce that the cutoff time can be written as a function only of $k$ and $|G|$ when $d(G) \ll \log |G|$ and $k - d(G) \asymp k \gg 1$; this is not the case when $d(G) \asymp \log |G| \asymp k$. For certain regimes of $k$, we find the limit profile of the convergence to equilibrium. Wilson (1997) conjectured that $\mathbb Z_2^d$ gives rise to the slowest mixing time for $G_k$ amongst all groups of size at most $2^d$. We give a partial answer, verifying the conjecture for nilpotent groups. This is obtained via a comparison result of independent interest between the mixing times of nilpotent $G$ and a corresponding Abelian group $\overline G$, namely the direct sum of the Abelian quotients in the lower central series of $G$. We use this to refine a celebrated result of Alon and Roichman (1994): we show for nilpotent $G$ that $G_k$ is an expander provided $k - d(\overline G) \gtrsim \log |G|$. As another consequence, we establish cutoff for nilpotent groups with relatively small commutators, including high-dimensional special groups, such as Heisenberg groups.
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".