Universal edge fluctuations of discrete interlaced particle systems
Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 1, pp. 75-197.

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 1 n, and examine the asymptotic behaviour, as n, 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 -1 3 ) and O(n -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.

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, 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 -1 3 ) et O(n -2 3 ), dans les directions respectivement tangentes et normales. Bien que les résultats de l’article soient naturels, les difficultés techniques liées à l’etude d’une si grande classe de modèles sous des hypothèses si faibles sont substantielles et inévitables.

Published online:
DOI: 10.5802/ambp.373
Classification: 60B20
Keywords: Random lozenge tilings, Universal edge fluctuations, Steepest descent
Erik Duse 1; Anthony Metcalfe 2

1 PB 68 Gustaf Hällströms gata 2b 000 14 Helsingfors, Finland
2 Kostervägen 2B 181 35 Lidingö, Sweden
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AMBP_2018__25_1_75_0,
     author = {Erik Duse and Anthony Metcalfe},
     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},
     volume = {25},
     number = {1},
     year = {2018},
     doi = {10.5802/ambp.373},
     mrnumber = {3851336},
     zbl = {1401.60010},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.373/}
}
TY  - JOUR
AU  - Erik Duse
AU  - Anthony Metcalfe
TI  - Universal edge fluctuations of discrete interlaced particle systems
JO  - Annales mathématiques Blaise Pascal
PY  - 2018
DA  - 2018///
SP  - 75
EP  - 197
VL  - 25
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.373/
UR  - https://www.ams.org/mathscinet-getitem?mr=3851336
UR  - https://zbmath.org/?q=an%3A1401.60010
UR  - https://doi.org/10.5802/ambp.373
DO  - 10.5802/ambp.373
LA  - en
ID  - AMBP_2018__25_1_75_0
ER  - 
%0 Journal Article
%A Erik Duse
%A Anthony Metcalfe
%T Universal edge fluctuations of discrete interlaced particle systems
%J Annales mathématiques Blaise Pascal
%D 2018
%P 75-197
%V 25
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.373
%R 10.5802/ambp.373
%G en
%F AMBP_2018__25_1_75_0
Erik Duse; Anthony Metcalfe. Universal edge fluctuations of discrete interlaced particle systems. Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 1, pp. 75-197. doi : 10.5802/ambp.373. https://ambp.centre-mersenne.org/articles/10.5802/ambp.373/

[1] Zhidong Bai; Jack W. Silverstein Spectral Analysis of Large Dimensional Random Matrices, Springer Series in Statistics, Springer, New York, 2010 | Zbl

[2] Alexei Borodin; Jeffrey Kuan Asymptotics of Plancherel measures for the infinite-dimensional unitary group, Adv. Math., Volume 219 (2008) no. 3, pp. 894-931 | DOI | MR | Zbl

[3] Alexey Bufetov; Knizel Knizel Asymptotics of random domino tilings of rectangular Aztec diamonds (2016) (https://arxiv.org/abs/1604.01491) | Zbl

[4] Sunil Chhita; Kurt Johansson; Benjamin Young Asymptotic domino statistics in the Aztec diamond, Ann. Appl. Probab., Volume 25 (2015) no. 3, pp. 1232-1278 | DOI | MR | Zbl

[5] Henry Cohn; Michael Larsen; James Propp The shape of a typical boxed plane partition, New York J. Math., Volume 4 (1998), pp. 137-165 | MR | Zbl

[6] Manon Defosseux Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 46 (2010) no. 1, pp. 209-249 | DOI | MR | Zbl

[7] Erik Duse; Kurt Johansson; Anthony Metcalfe The Cusp-Airy Process, Electron. J. Probab., Volume 21 (2016), 57, 50 pages | MR | Zbl

[8] Erik Duse; Anthony Metcalfe Asymptotic geometry of discrete interlaced patterns: Part I, Int. J. Math., Volume 26 (2015) no. 11, 1550093, 66 pages | MR | Zbl

[9] Erik Duse; Anthony Metcalfe Asymptotic geometry of discrete interlaced patterns: Part II (2015) (https://arxiv.org/abs/1507.00467) | Zbl

[10] Walid Hachem; Adrien Hardy; Jamal Najim Large Complex Correlated Wishart Matrices: Fluctuations and Asymptotic Independence at the Edges, Ann. Probab., Volume 44 (2016) no. 3, pp. 2264-2348 | DOI | MR | Zbl

[11] Kurt Johansson Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Commun. Math. Phys., Volume 215 (2001), pp. 683-705 | DOI | MR | Zbl

[12] Kurt Johansson Discrete polynuclear growth and determinantal processes, Commun. Math. Phys., Volume 242 (2003), pp. 277-329 | DOI | MR | Zbl

[13] Kurt Johansson The arctic circle boundary and the Airy process, Ann. Probab., Volume 33 (2005) no. 1, pp. 1-30 | MR | Zbl

[14] Kurt Johansson Random matrices and determinantal processes, Mathematical Statistical Physics, Session LXXXIII: Lecture Notes of the Les Houches Summer School, Elsevier, 2006, pp. 1-56 | MR | Zbl

[15] Kurt Johansson; Eric J.G. Nordenstam Eigenvalues of GUE Minors, Electron. J. Probab., Volume 11 (2006), pp. 1342-1371 (erratum in ibid., 12:1048–1051, 2007) | DOI | MR | Zbl

[16] Richard Kenyon; Andrei Okounkov Limit shapes and the complex Burgers equation, Acta Math., Volume 199 (2007) no. 2, pp. 263-302 | MR | Zbl

[17] Richard Kenyon; Andrei Okounkov; Scott Sheffield Dimers and Amoebae, Ann. Math., Volume 163 (2006) no. 3, pp. 1019-1056 | MR | Zbl

[18] Madan Lal Mehta Random Matrices, Pure and Applied Mathematics, 142, Elsevier, 2004 | MR | Zbl

[19] Anthony Metcalfe Universality properties of Gelfand-Tsetlin patterns, Probab. Theory Relat. Fields, Volume 155 (2013) no. 1-2, pp. 303-346 | DOI | MR | Zbl

[20] James Murray Asymptotic Analysis, Applied Mathematical Sciences, 48, Springer, New York, 1984 | MR | Zbl

[21] Leonid Pastur; Mariya Shcherbina Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Stat. Phys., Volume 86 (1997), pp. 109-147 | DOI | MR | Zbl

[22] Leonid Pastur; Mariya Shcherbina Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, 171, American Mathematical Society, 2011 | MR | Zbl

[23] Leonid Petrov Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Relat. Fields, Volume 160 (2014) no. 3-4, pp. 429-487 | DOI | MR | Zbl

[24] Michael Prähofer; Herbert Spohn Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys., Volume 108 (2002) no. 5-6, pp. 1071-1106 | DOI | MR | Zbl

[25] Craig A. Tracy; Harold Widom The Pearcey process, Commun. Math. Phys., Volume 263 (2006), pp. 381-400 | DOI | MR | Zbl

Cited by Sources: