Permutations with fixed pattern densities

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 sh...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 56; no. 1; pp. 220 - 250
Main Authors: Kenyon, Richard, Král', Daniel, Radin, Charles, Winkler, Peter
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.01.2020
Wiley Subscription Services, Inc
Subjects:
Constraints
Density
graphons
permutation patterns
Permutations
permutons
Phase transitions
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary: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.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20882