Cutoff for Random Walks on Upper Triangular Matrices
Bibliographic record
Abstract
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$ (ie $1 \ll k = |G|^{o(1)}$). A conjecture of Aldous and Diaconis (1985) asserts, for $k\gg\log|G|$, that the random walk on this graph exhibits cutoff. When $\log k \lesssim \log\log|G|$ (ie $k = (\log |G|)^{\mathcal O(1)}$), the only example of a non-Abelian group for which cutoff has been established is the dihedral group. We establish cutoff (as $p\to infty$) for the group of $d \times d$ unit upper triangular matrices with integer entries modulo $p$ (prime), which we denote $U_{p,d}$, for fixed $d$ or $d$ diverging sufficiently slowly. We allow $1 \ll k \lesssim \log |U_{p,d}|$ as well as $k\gg\log|U_{p,d}|$. The cutoff time is $\max\{\log_k |U_{p,d}|, \: s_0 k\}$, where $s_0$ is the time at which the entropy of the random walk on $\mathbb Z$ reaches $(\log |U_{p,d}^\mathrm{ab}|)/k$, where $U_{p,d}^\mathrm{ab} \cong \mathbb Z_p^{d-1}$ is the Abelianisation of $U_{p,d}$. When $1 \ll k \ll \log |U_{p,d}^\mathrm{ab}|$ and $d \asymp 1$, we find the limit profile. We also prove highly related results for the $d$-dimensional Heisenberg group over $\mathbb Z_p$. The Aldous--Diaconis conjecture also asserts, for $k gg\log |G|$, that the cutoff time should depend only on $k$ and $|G|$. This was verified for all Abelian groups. Our result shows that this is not the case for $U_{p,d}$: the cutoff time depends on $k$, $|U_{p,d}| = p^{d(d-1)/2}$ and $|U_{p,d}^\mathrm{ab}|=p^{d-1}$. We also show that all but $o(|U_{p,d}|)$ of the elements of $U_{p,d}$ lie at graph distance $M \pm o(M)$ from the identity, where $M$ is the minimal radius of a ball in $\mathbb Z^k$ of cardinality $|U_{p,d}^\mathrm{ab}| = p^{d-1}$. Finally, we show that the diameter is also asymptotically $M$ when $k \gtrsim \log |U_{p,d}^\textrm{ab}|$ and $d\asymp1$.
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How this classification was reachedexpand
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".