Abstract
The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an abc-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after appropriate identification of the anisotropy parameter therein.
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Baryshnikov Yu.: GUEs and queues. Probab. Theory Relat. Fields 119(2), 256–274 (2001)
Baik J., Borodin A., Deift P., Suidan T.: A model for the bus system in Cuernavaca (Mexico). J. Phys. A 39(28), 8965–8975 (2006)
Borodin A., Ferrari P.L., Prähofer M., Sasamoto T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129(5–6), 1055–1080 (2007)
Bosbach, C., Gawronski, W.: Strong asymptotics for Jacobi polynomials with varying weights. Methods Appl. Anal. 6(1), 39–54 (1999). Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part I
Boutillier C.: The bead model and limit behaviors of dimer models. Ann. Probab. 37(1), 107–142 (2009)
Borodin A., Rains E.M.: Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121(3–4), 291–317 (2005)
Chen L.-C., Ismail M.E.H.: On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22(5), 1442–1449 (1991)
Collins B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133(3), 315–344 (2005)
Forrester, P.J., Nagao, T.: Determinantal correlations for classical projection processes. (2008). arXiv:0801.0100
Forrester P.J., Nordenstam E.: The anti-symmetric GUE minor process. Mosc. Math. J. 9(4), 749–774, 934 (2009)
Forrester P.J., Nagao T., Honner G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Phys. B 553(3), 601–643 (1999)
Forrester, P.J.: Log-gases and random matrices. In: London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010)
Forrester P.J., Rains E.M.: Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter. Probab. Theory Relat. Fields 130(4), 518–576 (2004)
Forrester P.J., Rains E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131(1), 1–61 (2005)
Gorin V.E.: Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funktsional. Anal. i Prilozhen. 42(3), 23–44, 96 (2008)
Gawronski, W., Shawyer, B.: Strong asymptotics and the limit distribution of the zeros of Jacobi polynomials \({P_n^{(an+\alpha,bn+\beta)}}\). In: Progress in Approximation Theory, pp. 379–404. Academic Press, Boston (1991)
Izen S.H.: Refined estimates on the growth rate of Jacobi polynomials. J. Approx. Theory 144(1), 54–66 (2007)
Johansson K., Nordenstam E.: Eigenvalues of GUE minors. Electron. J. Probab. 11(50), 1342–1371 (2006)
Johansson K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123(2), 225–280 (2002)
Nagao T., Forrester P.J.: Multilevel dynamical correlation functions for Dyson’s Brownian motion model of random matrices. Phys. Lett. A 247(1–2), 42–46 (1998)
Nordenstam, E.: Interlaced particles in tilings and random matrices. PhD thesis, Swedish Royal Institute of Technology (KTH) (2009)
Okounkov A., Reshetikhin N.: The birth of a random matrix. Mosc. Math. J. 6(3), 553–566, 588 (2006)
Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975). American Mathematical Society, Colloquium Publications, vol. XXIII
Wachter K.W.: The limiting empirical measure of multiple discriminant ratios. Ann. Stat. 8(5), 937–957 (1980)
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Fleming, B.J., Forrester, P.J. & Nordenstam, E. A finitization of the bead process. Probab. Theory Relat. Fields 152, 321–356 (2012). https://doi.org/10.1007/s00440-010-0324-5
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DOI: https://doi.org/10.1007/s00440-010-0324-5