Universal edge fluctuations of discrete interlaced particle systems

Erik Duse; Anthony Metcalfe

doi:10.5802/ambp.373

Universal edge fluctuations of discrete interlaced particle systems
[Universalité au bord pour la fluctuation de systèmes discrets de particules entrelacées]

Erik Duse ¹ ; Anthony Metcalfe ²

¹ PB 68 Gustaf Hällströms gata 2b 000 14 Helsingfors, Finland
² Kostervägen 2B 181 35 Lidingö, Sweden

Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 1, pp. 75-197.

Résumé
Abstract

Nous considérons la mesure uniforme sur l’ensemble des configurations de Gelfand–Tsetlin de profondeur $n$ après avoir fixé la position des particules de la n-ième ligne. De maniere équivalente, ces systèmes décrivent une grande classe de modèles de pavages aléatoires et ont un rapport étroit avec les processus de valeurs propres de mineurs d’une grande classe de matrices aléatoires hermitiennes. Ils ont une structure déterminantale et leur noyau de corrélation est connu. Nous redimensionnons le système par un facteur $1 / n$ , et examinons son comportement asymptotique lorsque $n \to \infty$ , sous l’hypothèse faible pour les particules sur la rangée $n$ , que la distribution empirique redimensionnée de ces dernières converge faiblement vers une mesure de probabilité avec support compact, et que cette dernière satisfasse un minimum de régularité.

Nous prouvons que le noyau de corrélation des particules dans le voisinage d’un « point typique du bord » converge vers le noyau de Airy étendu. A cette fin, nous trouvons dans un premier temps un dimensionnement adéquat pour la fluctuation des particules. Nous donnons une paramétrisation explicite du noyau asymptotique, définissons une courbe non-asymptotique analogue (et son équivalent en dimension $n$ ), et choisissons notre scaling de telle sorte que les particules fluctuent autour de cette courbe avec des ordres $O (n^{- \frac{1}{3}})$ et $O (n^{- \frac{2}{3}})$ , dans les directions respectivement tangentes et normales. Bien que les résultats de l’article soient naturels, les difficultés techniques liées Ã l’etude d’une si grande classe de modèles sous des hypothèses si faibles sont substantielles et inévitables.

We impose the uniform probability measure on the set of all discrete Gelfand–Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac{1}{n}$ , and examine the asymptotic behaviour, as $n \to \infty$ , under weak asymptotic assumptions for the (rescaled) particles on row $n$ : The empirical distribution of these converges weakly to a probability measure with compact support, and they otherwise satisfy mild regulatory restrictions.

We prove that the correlation kernel of particles in the neighbourhood of “typical edge points” convergences to the extended Airy kernel. To do this, we first find an appropriate scaling for the fluctuations of the particles. We give an explicit parameterisation of the asymptotic edge, define an analogous non-asymptotic edge curve (or finite $n$ -deterministic equivalent), and choose our scaling such that the particles fluctuate around this with fluctuations of order $O (n^{- \frac{1}{3}})$ and $O (n^{- \frac{2}{3}})$ in the tangent and normal directions respectively. While the final results are quite natural, the technicalities involved in studying such a broad class of models under such weak asymptotic assumptions are unavoidable and extensive.

Publié le : 2018-07-01

MR Zbl

DOI : 10.5802/ambp.373

Classification : 60B20
Keywords: Random lozenge tilings, Universal edge fluctuations, Steepest descent
Mot clés : Random lozenge tilings, Universal edge fluctuations, Steepest descent

Affiliations des auteurs :

Erik Duse ¹ ; Anthony Metcalfe ²

¹ PB 68 Gustaf Hällströms gata 2b 000 14 Helsingfors, Finland
² Kostervägen 2B 181 35 Lidingö, Sweden

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Universal edge fluctuations of discrete interlaced particle systems},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {75--197},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Erik Duse; Anthony Metcalfe. Universal edge fluctuations of discrete interlaced particle systems. Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 1, pp. 75-197. doi : 10.5802/ambp.373. https://ambp.centre-mersenne.org/articles/10.5802/ambp.373/

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