## The Annals of Probability

### On global fluctuations for non-colliding processes

Maurice Duits

#### Abstract

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

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

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

https://projecteuclid.org/euclid.aop/1523520018

Digital Object Identifier
doi:10.1214/17-AOP1185

Mathematical Reviews number (MathSciNet)
MR3785589

Zentralblatt MATH identifier
06894775

#### Citation

Duits, Maurice. On global fluctuations for non-colliding processes. Ann. Probab. 46 (2018), no. 3, 1279--1350. doi:10.1214/17-AOP1185. https://projecteuclid.org/euclid.aop/1523520018

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