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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.

MSC:
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
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