The Annals of Probability

On global fluctuations for non-colliding processes

Maurice Duits

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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.

Article information

Ann. Probab., Volume 46, Number 3 (2018), 1279-1350.

Received: April 2016
Revised: March 2017
First available in Project Euclid: 12 April 2018

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Non-colliding processes Gaussian Free Field Central Limit Theorems determinantal point processes orthogonal polynomials


Duits, Maurice. On global fluctuations for non-colliding processes. Ann. Probab. 46 (2018), no. 3, 1279--1350. doi:10.1214/17-AOP1185.

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  • [1] Akemann, G., Baik, J. and Di Francesco, P., eds. (2011). The Oxford Handbook of Random Matrix Theory. Oxford Univ. Press, Oxford.
  • [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.
  • [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.
  • [4] Borodin, A. (1999). Biorthogonal ensembles. Nuclear Phys. B 536 704–732.
  • [5] Borodin, A. (2011). Determinantal point processes. In The Oxford Handbook of Random Matrix Theory 231–249. Oxford Univ. Press, Oxford.
  • [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.
  • [7] Borodin, A. and Bufetov, A. (2014). Plancherel representations of $U(\infty)$ and correlated Gaussian free fields. Duke Math. J. 163 2109–2158.
  • [8] Borodin, A. and Ferrari, P. L. (2014). Anisotropic growth of random surfaces in $2+1$ dimensions. Comm. Math. Phys. 325 603–684.
  • [9] Borodin, A. and Gorin, V. (2015). General $\beta$-Jacobi corners process and the Gaussian free field. Comm. Pure Appl. Math. 68 1774–1844.
  • [10] Borodin, A. and Olshanski, G. (2006). Markov processes on partitions. Probab. Theory Related Fields 135 84–152.
  • [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.
  • [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.
  • [14] Bufetov, A. and Gorin, V. Fluctuations of particle systems determined by Schur generating functions. ArXiv preprint. Available at arXiv:1604.01110.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [22] Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191–1198.
  • [23] Ehrhardt, T. (2003). A generalization of Pincus’ formula and Toeplitz operator determinants. Arch. Math. (Basel) 80 302–309.
  • [24] Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators. Operator Theory: Advances and Applications 116. Birkhäuser, Basel.
  • [25] Gorin, V. E. (2008). Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funktsional. Anal. i Prilozhen. 42 23–44, 96.
  • [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.
  • [28] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204.
  • [29] Johansson, K. (2005). The Arctic circle boundary and the Airy process. Ann. Probab. 33 1–30.
  • [30] Johansson, K. (2005). Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier (Grenoble) 55 2129–2145.
  • [31] Johansson, K. (2006). Random matrices and determinantal processes. In Mathematical Statistical Physics 1–55. Elsevier B. V., Amsterdam.
  • [32] Kenyon, R. (2008). Height fluctuations in the honeycomb dimer model. Comm. Math. Phys. 281 675–709.
  • [33] Koekoek, R., Lesky, P. A. and Swarttouw, R. F. (2010). Hypergeometric Orthogonal Polynomials and Their $q$-Analogues. Springer Monographs in Mathematics. Springer, Berlin.
  • [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.
  • [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.
  • [39] Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98 167–212.
  • [40] O’Connell, N. (2003). Conditioned random walks and the RSK correspondence. J. Phys. A 36 3049–3066.
  • [41] O’Connell, N. and Yor, M. (2002). A representation for non-colliding random walks. Electron. Commun. Probab. 7 1–12.
  • [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.
  • [44] Schoutens, W. (2000). Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics 146. Springer, New York.
  • [45] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [46] Simon, B. (2005). Trace Ideals and Their Applications, 2nd ed. Mathematical Surveys and Monographs 120. Amer. Math. Soc., Providence, RI.
  • [47] Simon, B. (2011). Szegős Theorem and Its Descendants: Spectral Theory for $L^{2}$ Perturbations of Orthogonal Polynomials. Princeton Univ. Press, Princeton, NJ.
  • [48] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107–160.
  • [49] Soshnikov, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab. 30 171–187.
  • [50] Van Assche, W. and Coussement, E. (2001). Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127 317–347.