{"paper":{"title":"A log-Sobolev inequality for the multislice, with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Ryan O'Donnell, Xinyu Wu, Yuval Filmus","submitted_at":"2018-09-10T18:46:07Z","abstract_excerpt":"Let $\\kappa \\in \\mathbb{N}_+^\\ell$ satisfy $\\kappa_1 + \\dots + \\kappa_\\ell = n$ and let $\\mathcal{U}_\\kappa$ denote the \"multislice\" of all strings $u$ in $[\\ell]^n$ having exactly $\\kappa_i$ coordinates equal to $i$, for all $i \\in [\\ell]$. Consider the Markov chain on $\\mathcal{U}_\\kappa$, where a step is a random transposition of two coordinates of $u$. We show that the log-Sobolev constant $\\rho_\\kappa$ for the chain satisfies $$(\\rho_\\kappa)^{-1} \\leq n \\sum_{i=1}^{\\ell} \\tfrac{1}{2} \\log_2(4n/\\kappa_i),$$ which is sharp up to constants whenever $\\ell$ is constant. From this, we derive so"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.03546","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}