Increasing subsequences of random walks
Bibliographic record
Abstract
Abstract Given a sequence of n real numbers { S i } i ⩽ n , we consider the longest weakly increasing subsequence, namely i 1 < i 2 < . . . < i L with S i k ⩽ S i k+1 and L maximal. When the elements S i are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that ${\mathbb E} L=(2+o(1)) \sqrt{n}$ . We consider the case when { S i } i⩽n is a random walk on ℝ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies ${\mathbb E} L\geq c\sqrt{n}$ . Our main result is an upper bound ${\mathbb E} L\leq n^{1/2 + o(1)}$ , establishing the leading asymptotic behavior. If { S i } i⩽n is a simple random walk on ℤ, we improve the lower bound by showing that ${\mathbb E} L \geq c\sqrt{n} \log{n}$ . We also show that if { S i } is a simple random walk in ℤ 2 , then there is a subsequence of { S i } i⩽n of expected length at least cn 1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n 1/2+ o (1) . The problem of determining the correct exponent remains open.
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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 | 0.007 | 0.012 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| 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".