Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The local limit of random sorting networks

Omer Angel, Duncan Dauvergne, Alexander E. Holroyd, and Bálint Virág

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A sorting network is a geodesic path from $12\cdots n$ to $n\cdots21$ 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 $a\in [0,1]$ as $n\to \infty $. Here time is scaled by a factor of $1/n$ and space is not scaled.

The limit is a swap process $U$ on $\mathbb{Z}$. 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 $\sqrt{a(1-a)}$.

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.


Un réseau de tri est un chemin géodésique de $12\cdots n$ à $n\cdots 21$ dans le graphe de Cayley de $S_{n}$ généré par les transpositions adjacentes. Pour un réseau de tri uniforme, on établit l’existence d’une limite locale du processus des positions espace-temps des transpositions dans un voisinage de $an$ pour $a\in [0,1]$ lorsque $n\to \infty $. Ici, le temps est mis à l’échelle par un facteur de $1/n$ et l’espace n’est pas mis à l’échelle.

La limite est un processus d’échange $U$ sur $\mathbb{Z}$. On montre que $U$ est stationnaire et mélangeant par rapport au déplacement spatial, et qu’il a des incréments de temps qui sont stationnaires. De plus, la seule dépendance sur $a$ est à travers une mise à l’échelle temporelle par un facteur de $\sqrt{a(1-a)}$.

Pour établir l’existence de $U$, on trouve une limite locale pour les tableaux de Young en forme d’escalier. Ces tableaux de Young sont reliés aux réseaux de tri à travers une bijection d’Edelman et Greene.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 412-440.

Received: 20 April 2017
Revised: 2 November 2017
Accepted: 23 January 2018
First available in Project Euclid: 18 January 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05E10: Combinatorial aspects of representation theory [See also 20C30] 68P10: Searching and sorting

Sorting network Random sorting network Reduced decomposition Young tableau Local limit


Angel, Omer; Dauvergne, Duncan; Holroyd, Alexander E.; Virág, Bálint. The local limit of random sorting networks. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 412--440. doi:10.1214/18-AIHP887.

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