Random walks on regular trees can not be slowed down
Bibliographic record
Abstract
We study a permuted random walk process on a graph G. Given a fixed sequence of permutations on the vertices of G, the permuted random walker alternates between taking random walk steps, and applying the next permutation in the sequence to their current position. Existing work on permuted random walks includes results on hitting times, mixing times, and asymptotic speed. The usual random walk on a regular tree, or generally any non-amenable graph, has positive speed, i.e. the distance from the origin grows linearly. Our focus is understanding whether permuted walks can be slower than the corresponding non-permuted walk, by carefully choosing the permutation sequence. We show that on regular trees (including the line), the permuted random walk is always stochastically faster. The proof relies on a majorization inequality for probability measures, plus an isoperimetric inequality for the tree. We also quantify how much slower the permuted random walk can possibly be when it is coupled with the corresponding non-permuted walk.
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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 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".