Open Access
November 2020 A covariance formula for topological events of smooth Gaussian fields
Dmitry Beliaev, Stephen Muirhead, Alejandro Rivera
Ann. Probab. 48(6): 2845-2893 (November 2020). DOI: 10.1214/20-AOP1438


We derive a covariance formula for the class of ‘topological events’ of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 1679–1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.


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Dmitry Beliaev. Stephen Muirhead. Alejandro Rivera. "A covariance formula for topological events of smooth Gaussian fields." Ann. Probab. 48 (6) 2845 - 2893, November 2020.


Received: 1 December 2018; Revised: 1 March 2020; Published: November 2020
First available in Project Euclid: 20 October 2020

MathSciNet: MR4164455
Digital Object Identifier: 10.1214/20-AOP1438

Primary: 60G60
Secondary: 60D05 , 60G15

Keywords: covariance formula , Gaussian fields , topology

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 6 • November 2020
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