Record-biased permutations and their permuton limit

TL;DR

This paper investigates record-biased permutations, providing generative models, expectation calculations, limit distributions, and establishing convergence to a deterministic permuton as bias increases.

Contribution

It introduces new generative processes for record-biased permutations and characterizes their permuton limit in the high-bias regime.

Findings

Derived expectations for classical permutation statistics.

Obtained limit distributions in the logarithmic record regime.

Proved convergence to a deterministic permuton at high bias.

Abstract

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call \emph{record-biased permutations}. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterize. This model was introduced in our earlier work [N. Auger, M. Bouvel, C. Nicaud, C. Pivoteau, \emph{Analysis of Algorithms for Permutations Biased by Their Number of Records}, AofA 2016], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.