ON GLOBAL FLUCTUATIONS FOR NON-COLLIDING PROCESSES

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 c...

Full description

Saved in:
Bibliographic Details
Published in:The Annals of probability 2018-05, Vol.46 (3), p.1279-1350
Main Author: Duits, Maurice
Format: Article
Language:English
Subjects:
Asymptotic methods
Asymptotic properties
Central Limit Theorems
determinantal point processes
Gaussian Free Field
Globalization
Multilayers
Non-colliding processes
Normal distribution
orthogonal polynomials
Studies
Variation
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
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.
ISSN:0091-1798
2168-894X
2168-894X
DOI:10.1214/17-AOP1185