{"id":"openai-math/Random-coordinate-frames-and-partial-permutation-laws","pdf":"https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026","tags":["probability","mixing-times","card-shuffling","combinatorics"],"score":0,"title":"Random coordinate frames and partial permutation laws","domain":"Probability and statistical mechanics","family":"238","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"For the Thorp shuffle on $`2^d`$ cards, we prove that the worst-start total-variation distance after $`32800d`$ physical shuffles tends to zero as $`d\\to\\infty`$. Combining our fixed-list estimate with the companion Fourier transfer improves this bound to $`512d`$ shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":false,"family_title":"Optimal logarithmic mixing of the Thorp shuffle"}