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May 2018 On global fluctuations for non-colliding processes
Maurice Duits
Ann. Probab. 46(3): 1279-1350 (May 2018). DOI: 10.1214/17-AOP1185


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.


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Maurice Duits. "On global fluctuations for non-colliding processes." Ann. Probab. 46 (3) 1279 - 1350, May 2018.


Received: 1 April 2016; Revised: 1 March 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894775
MathSciNet: MR3785589
Digital Object Identifier: 10.1214/17-AOP1185

Primary: 42C05 , 60F05 , 60K35

Keywords: central limit theorems , Determinantal point processes , Gaussian free field , Non-colliding processes , orthogonal polynomials

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • May 2018
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