The Archimedean limit of random sorting networks
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Bibliographic record
Abstract
A sorting network (also known as a reduced decomposition of the reverse permutation) is a shortest path from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="12 midline-horizontal-ellipsis n"> <mml:semantics> <mml:mrow> <mml:mn>12</mml:mn> <mml:mo> ⋯ </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">12 \cdots n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n midline-horizontal-ellipsis 21"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ⋯ </mml:mo> <mml:mn>21</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \cdots 21</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Cayley graph of the symmetric group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generated by adjacent transpositions. We prove that in a uniform random <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -element sorting network <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript n"> <mml:semantics> <mml:msup> <mml:mi> σ </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> permutation matrix measures of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript n"> <mml:semantics> <mml:msup> <mml:mi> σ </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As a corollary of these results, we show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is embedded into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via the map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau right-arrow from bar left-parenthesis tau left-parenthesis 1 right-parenthesis comma tau left-parenthesis 2 right-parenthesis comma ellipsis tau left-parenthesis n right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> τ </mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau \mapsto (\tau (1), \tau (2), \dots \tau (n))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then with high probability, the path <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript n"> <mml:semantics> <mml:msup> <mml:mi> σ </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is close to a great circle on a particular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n minus 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − <
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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 | 0.006 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.003 | 0.005 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.003 |
| 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