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Permutations with fixed pattern densities
Author(s) -
Kenyon Richard,
Král' Daniel,
Radin Charles,
Winkler Peter
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20882
Subject(s) - fixed point , limit (mathematics) , statistical physics , scaling , mathematics , combinatorics , physics , mathematical analysis , geometry
We study scaling limits of random permutations (“permutons”) constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed 12 density, with fixed 12 and 123 densities, with fixed 12 density and the sum of 123 and 213 densities, and with fixed 123 and 321 densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.