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Monotone subsequences in locally uniform random permutations

Sjöstrand, Jonas

The Annals of probability, 2023-07, Vol.51 (4), p.1502 [Periódico revisado por pares]

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Título:
Monotone subsequences in locally uniform random permutations
Autor: Sjöstrand, Jonas
Assuntos: decreasing subsequence ; increasing subsequence ; limit shape ; Random permutation ; Robinson–Schensted ; Young diagram
É parte de: The Annals of probability, 2023-07, Vol.51 (4), p.1502
Descrição: A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the j th point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any r ≥ 0, the largest union of (Formula Presenmted)decreasing subsequences approaches a limit surface as n tends to infinity, and the limit surface is a solution to a specific variational problem. As a corollary, we prove the existence of a limit shape for the Young diagram associated to the random permutation under the Robinson– Schensted correspondence. In the special case where ρ is the uniform distribution on the diamond |x| + |y| < 1, we conjecture that the limit shape is triangular, and assuming the conjecture is true, we find an explicit formula for the limit surfaces of a uniformly random permutation and recover the famous limit shape of Vershik, Kerov and Logan, Shepp.
Idioma: Inglês

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Monotone subsequences in locally uniform random permutations

Sjöstrand, Jonas

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