{"id":"openai-math/Row-column-symmetry-and-contraction-of-coordinate-sweeps","pdf":"https://github.com/openai/math/blob/main/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026","tags":["probability","markov-chains","card-shuffling"],"score":0,"title":"Row–column symmetry and contraction of coordinate sweeps","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 $`N=2^d`$ labeled cards has full-permutation total-variation mixing time $`\\Theta(\\log N)`$ in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.","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"}