{"id":"openai-math/A-strict-four-row-permanent-inequality-and-permutation-moments","pdf":"https://github.com/openai/math/blob/main/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026","tags":["combinatorics","permanents","card-shuffling"],"score":0,"title":"A strict four-row permanent inequality and permutation moments","domain":"Probability and statistical mechanics","family":"238","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove a permanent inequality with exponent strictly below two for laws on S 4 sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on $`N=2^d`$ cards to total-variation distance tending to zero from uniform as $`N\\to\\infty`$. Thus the mixing time has the optimal order $`O(\\log N)`$ in physical shuffles.","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"}