zbMATH — the first resource for mathematics

On global fluctuations for non-colliding processes. (English) Zbl 1429.60072
Summary: We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove central limit theorems for the linear statistics. We then show that these results prove Gaussian free field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
Full Text: DOI Euclid arXiv
[1] Akemann, G., Baik, J. and Di Francesco, P., eds. (2011). The Oxford Handbook of Random Matrix Theory. Oxford Univ. Press, Oxford. · Zbl 1225.15004
[2] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge. · Zbl 1184.15023
[3] Bleher, P. M. and Kuijlaars, A. B. J. (2005). Integral representations for multiple Hermite and multiple Laguerre polynomials. Ann. Inst. Fourier (Grenoble) 55 2001-2014. · Zbl 1084.33008
[4] Borodin, A. (1999). Biorthogonal ensembles. Nuclear Phys. B 536 704-732. · Zbl 0948.82018
[5] Borodin, A. (2011). Determinantal point processes. In The Oxford Handbook of Random Matrix Theory 231-249. Oxford Univ. Press, Oxford. · Zbl 1238.60055
[6] Borodin, A. (2014). CLT for spectra of submatrices of Wigner random matrices, II: Stochastic evolution. In Random Matrix Theory, Interacting Particle Systems, and Integrable Systems. Math. Sci. Res. Inst. Publ.65 57-69. Cambridge Univ. Press, New York. · Zbl 1330.60013
[7] Borodin, A. and Bufetov, A. (2014). Plancherel representations of \(U(∞)\) and correlated Gaussian free fields. Duke Math. J.163 2109-2158. · Zbl 1322.60062
[8] Borodin, A. and Ferrari, P. L. (2014). Anisotropic growth of random surfaces in \(2+1\) dimensions. Comm. Math. Phys.325 603-684. · Zbl 1303.82015
[9] Borodin, A. and Gorin, V. (2015). General \(β\)-Jacobi corners process and the Gaussian free field. Comm. Pure Appl. Math.68 1774-1844. · Zbl 1325.60076
[10] Borodin, A. and Olshanski, G. (2006). Markov processes on partitions. Probab. Theory Related Fields 135 84-152. · Zbl 1105.60052
[11] Böttcher, A. and Silbermann, B. (1999). Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer, New York.
[12] Breuer, J. and Duits, M. (2016). Universality of mesoscopic fluctuations for orthogonal polynomial ensembles. Comm. Math. Phys.342 491-531. · Zbl 1360.42014
[13] Breuer, J. and Duits, M. (2017). Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. J. Amer. Math. Soc.30 27-66. · Zbl 1351.15022
[14] Bufetov, A. and Gorin, V. Fluctuations of particle systems determined by Schur generating functions. ArXiv preprint. Available at arXiv:1604.01110. · Zbl 1400.82064
[15] Costin, O. and Lebowitz, J. L. (1995). Gaussian fluctuation in random matrices. Phys. Rev. Lett.75 69-72.
[16] Daems, E. and Kuijlaars, A. B. J. (2007). Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions. J. Approx. Theory 146 91-114. · Zbl 1135.42324
[17] Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X. (1999). Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in Random Matrix Theory. Comm. Pure Appl. Math.52 1335-1425. · Zbl 0944.42013
[18] Doumerc, Y. (2005). Matrices aléatoires, processus stochastiques et groupes de réflexions Ph.D. thesis.
[19] Duits, M. (2013). Gaussian free field in an interlacing particle system with two jump rates. Comm. Pure Appl. Math.66 600-643. · Zbl 1259.82091
[20] Duits, M., Geudens, D. and Kuijlaars, A. B. J. (2011). A vector equilibrium problem for the two-matrix model in the quartic/quadratic case. Nonlinearity 24 951-993. · Zbl 1211.31006
[21] Duits, M., Kuijlaars, A. B. J. and Mo, M. Y. (2012). The Hermitian two matrix model with an even quartic potential. Mem. Amer. Math. Soc.217 v+105. · Zbl 1247.15032
[22] Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys.3 1191-1198. · Zbl 0111.32703
[23] Ehrhardt, T. (2003). A generalization of Pincus’ formula and Toeplitz operator determinants. Arch. Math. (Basel) 80 302-309. · Zbl 1042.47013
[24] Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators. Operator Theory: Advances and Applications 116. Birkhäuser, Basel. · Zbl 0946.47013
[25] Gorin, V. E. (2008). Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funktsional. Anal. i Prilozhen.42 23-44, 96. · Zbl 1177.60012
[26] Gorin, V. E. (2008). Noncolliding Jacobi diffusions as the limit of Markov chains on the Gelfand-Tsetlin graph. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 360 91-123, 296.
[27] Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129. Cambridge Univ. Press, Cambridge. · Zbl 0887.60009
[28] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J.91 151-204. · Zbl 1039.82504
[29] Johansson, K. (2005). The Arctic circle boundary and the Airy process. Ann. Probab.33 1-30. · Zbl 1096.60039
[30] Johansson, K. (2005). Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier (Grenoble) 55 2129-2145. · Zbl 1083.60079
[31] Johansson, K. (2006). Random matrices and determinantal processes. In Mathematical Statistical Physics 1-55. Elsevier B. V., Amsterdam. · Zbl 1411.60144
[32] Kenyon, R. (2008). Height fluctuations in the honeycomb dimer model. Comm. Math. Phys.281 675-709. · Zbl 1157.82028
[33] Koekoek, R., Lesky, P. A. and Swarttouw, R. F. (2010). Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer Monographs in Mathematics. Springer, Berlin. · Zbl 1200.33012
[34] König, W. (2005). Orthogonal polynomial ensembles in probability theory. Probab. Surv.2 385-447.
[35] König, W. and O’Connell, N. (2001). Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Commun. Probab.6 107-114.
[36] König, W., O’Connell, N. and Roch, S. (2002). Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles. Electron. J. Probab.7 no. 5, 24.
[37] Kuan, J. (2014). The Gaussian free field in interlacing particle systems. Electron. J. Probab.19 no. 72, 31. · Zbl 1315.60061
[38] Kuijlaars, A. B. J. (2010). Multiple orthogonal polynomial ensembles. In Recent Trends in Orthogonal Polynomials and Approximation Theory. Contemp. Math.507 155-176. Amer. Math. Soc., Providence, RI. · Zbl 1218.60005
[39] Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci.98 167-212. · Zbl 1055.60003
[40] O’Connell, N. (2003). Conditioned random walks and the RSK correspondence. J. Phys. A 36 3049-3066. · Zbl 1035.05097
[41] O’Connell, N. and Yor, M. (2002). A representation for non-colliding random walks. Electron. Commun. Probab.7 1-12. · Zbl 1037.15019
[42] Olshanski, G. (2010). Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 378 81-110, 230.
[43] Petrov, L. (2015). Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. Ann. Probab.43 1-43. · Zbl 1315.60062
[44] Schoutens, W. (2000). Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics 146. Springer, New York. · Zbl 0960.60076
[45] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521-541. · Zbl 1132.60072
[46] Simon, B. (2005). Trace Ideals and Their Applications, 2nd ed. Mathematical Surveys and Monographs 120. Amer. Math. Soc., Providence, RI. · Zbl 1074.47001
[47] Simon, B. (2011). Szegős Theorem and Its Descendants: Spectral Theory for \(L^{2}\) Perturbations of Orthogonal Polynomials. Princeton Univ. Press, Princeton, NJ. · Zbl 1230.33001
[48] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107-160. · Zbl 0991.60038
[49] Soshnikov, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab.30 171-187. · Zbl 1033.60063
[50] Van Assche, W. and Coussement, E. (2001). Some classical multiple orthogonal polynomials. J. Comput. Appl. Math.127 317-347. · Zbl 0969.33005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.