### Abstract

A sorting network is a geodesic path from 12 · · · n to n · · · 21 in the Cayley graph of S_{n} generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space–time locations of transpositions in a neighbourhood of an for a ∈ [0, 1] as n → ∞. Here time is scaled by a factor of 1/n and space is not scaled. The limit is a swap process U on Z. We show that U is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on a is through time scaling by a factor of a(1 − a). To establish the existence of U, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.

Original language | English (US) |
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Pages (from-to) | 412-440 |

Number of pages | 29 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2019 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Local limit
- Random sorting network
- Reduced decomposition
- Sorting network
- Young tableau

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## Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*55*(1), 412-440. https://doi.org/10.1214/18-AIHP887