Cutoff for Almost All Random Walks on Abelian Groups
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it