{"id":"openai-math/Optimal-order-mixing-of-the-Thorp-shuffle","pdf":"https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026","tags":["probability","markov-chains","card-shuffling"],"score":0,"title":"Optimal-order mixing of the Thorp shuffle","domain":"Probability and statistical mechanics","family":"238","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove that the Thorp shuffle on $`2^d`$ cards mixes in $`\\Theta(d)`$ complete shuffles. The full permutation law after $`1600d`$ shuffles converges to uniform in total variation as $`d\\to\\infty`$, uniformly over the initial deck, while a support count gives a lower bound of $`2d-O(1)`$.","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"}