The RSK and longest-increasing-subsequence consequences remain owned by the hook-length/RSK page

This page consumes the distribution of the Robinson-Schensted shape of a uniform permutation (The RSK shape of a uniform random permutation has the Plancherel law) and Schensted's longest increasing and decreasing subsequence theorem only to localize Plancherel profiles (The RSK union bound localizes Plancherel profiles). It does not re-mint RSK, the LIS/LDS identities, the Baik-Deift-Johansson theorem, the Tracy-Widom distribution, determinantal point processes or edge statistics of Plancherel measure: those belong to the hook-length/RSK page and to separate analytic-probability suppliers, and the limit-shape theorem proved here (Plancherel Young diagrams converge to the limit shape) is the qualitative law of large numbers, not a fluctuation or edge result. In particular no sharp constant, rate or fluctuation-distribution statement is asserted: the localization lemma gives only the bound ${\lambda }_1,{\lambda }_1^{\mathrm{\prime }}\le C\sqrt{n}$ with probability tending to one, for each fixed $C>e$, and the limit-shape theorem gives only convergence in probability of the scaled profile to $\mathrm{\Omega }$.