Non-intersecting, simple, symmetric random walks and the extended Hahn kernel  [ Marches aléatoires simples, symétriques et qui ne s’intersectent pas et le noyau de Hahn étendu ]
Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2129-2145.

Nous montrons en utilisant des chemins qui ne s’intersectent pas qu’un pavage rhombique d’un hexagone, ou une partition planaire en boîtes, est décrit par un point processus ponctuel déterminentiel, donné par un noyau de Hahn étendu.

We show using non-intersecting paths, that a random rhombus tiling of a hexagon, or a boxed planar partition, is described by a determinantal point process given by an extended Hahn kernel.

DOI : https://doi.org/10.5802/aif.2155
Classification : 60K35,  15A32
Mots clés: chemins qui ne s’intersectent pas, mouvement brownien de Dyson, partitions planaires, pavages aléatoires, processus déterminentiels
@article{AIF_2005__55_6_2129_0,
     author = {Johansson, Kurt},
     title = {Non-intersecting, simple, symmetric  random walks and the extended Hahn kernel},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     pages = {2129-2145},
     doi = {10.5802/aif.2155},
     mrnumber = {2187949},
     zbl = {1083.60079},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2005__55_6_2129_0/}
}
Johansson, Kurt. Non-intersecting, simple, symmetric  random walks and the extended Hahn kernel. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2129-2145. doi : 10.5802/aif.2155. https://aif.centre-mersenne.org/item/AIF_2005__55_6_2129_0/

[1] G.E. Andrews; R. Askey; R. Roy Special Functions, Encyclopedia of Mathematics and its applications, Tome 71, Cambridge University Press, Cambridge, 1999 | MR 1688958 | Zbl 0920.33001

[2] J. Baik; T. Kriecherbauer; K.D.T.-R MacLaughlin; P. Miller Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles (math.CA/0310278, http://arxiv.org/abs/math.CA/0310278)

[3] H. Cohn; M. Larsen; J. Propp The shape of a typical boxed plane partition, New York J. of Math., Tome 4 (1998), pp. 137-165 | MR 1641839 | Zbl 0908.60083

[4] F. J. Dyson A Brownian-Motion Model for the eigenvalues of a Random Matrix, J. Math. Phys., Tome 3 (1962), pp. 1191-1198 | Article | MR 148397 | Zbl 0111.32703

[5] B. Eynard; M. L. Mehta Matrices coupled in a chain I: Eigenvalue correlations, J. of Phys. A, Tome 31 (1998), pp. 4449-4456 | Article | MR 1628667 | Zbl 0938.15012

[6] P. L. Ferrari; H. Spohn Step fluctuations for a faceted crystal, J. Stat. Phys., Tome 113 (2003), pp. 1-46 | Article | MR 2012974 | Zbl 02002354

[7] P. J. Forrester; T. Nagao; G. Honner Correlations for the orthogonal-unitary and symplectic-unitary transitions at the soft and hard edges, Nucl. Phys. B, Tome 553 (1999), pp. 601-643 | Article | MR 1707162 | Zbl 0944.82012

[8] K. Holmaker On a discrete Rodrigues' formula and a second class of orthogonal Hahn polynomials (Preprint, Department of Mathematics, Chalmers University of Technology, N° 1977-12)

[9] K. Johansson Discrete orthogonal polynomial ensembles and the Plancherel measure, Annals of Math., Tome 153 (2001), pp. 259-296 | Article | MR 1826414 | Zbl 0984.15020

[10] K. Johansson Non-intersecting paths, random tilings and random matrices, Probab.Theory Relat. Fields, Tome 123 (2002), pp. 225-280 | Article | MR 1900323 | Zbl 1008.60019

[11] K. Johansson Discrete polynuclear growth and determinantal processes, Commun. Math. Phys., Tome 242 (2003), pp. 277-329 | MR 2018275 | Zbl 1031.60084

[12] K. Johansson The Arctic circle and the Airy process (math.PR/0306216, to appear in Ann. Probab., http://arxiv.org/abs/math.PR/0306216) | MR 2118857 | Zbl 1096.60039

[13] M. Katori; H. Tanemura Scaling limit of vicious walks and two-matrix model, Phys. Rev. E (2002)

[14] R. Kenyon Local statistics of lattice dimers, Ann. Inst. H. Poincaré, Probabilités et Statistiques, Tome 33 (1997), pp. 591-618 | Article | Numdam | MR 1473567 | Zbl 0893.60047

[15] M. L. Mehta Random Matrices, 2nd ed., Academic Press, San Diego, 1991 | MR 1083764 | Zbl 0780.60014

[16] A. F. Nikiforov; S. K. Suslov; V. B. Uvarov Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Tome Berlin Heidelberg, Berlin Heidelberg, 1991 | MR 1149380 | Zbl 0576.33001

[17] M. Prähofer; H. Spohn Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys., Tome 108 (2002), pp. 1076-1106 | MR 1933446 | Zbl 1025.82010

[18] R. P. Stanley Enumerative Combinatorics, Cambridge University Press, Tome 2 (1999) | Zbl 0928.05001

[19] J. R. Stembridge Nonintersecting Paths, Pfaffians, and Plane Partitions, Adv. in Math., Tome 83 (1990), pp. 96-131 | Article | MR 1069389 | Zbl 0790.05007