Universal edge fluctuations of discrete interlaced particle systems

Duse, Erik; Metcalfe, Anthony

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

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 $O\left(n^{-\frac{1}{3}}\right)$ et $O\left(n^{-\frac{2}{3}}\right)$, 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.

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 $O\left(n^{-\frac{1}{3}}\right)$ and $O\left(n^{-\frac{2}{3}}\right)$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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