Geometry of smooth Gaussian fields

16 Nov 2020, 12:00 → 23 Nov 2020, 17:00 Europe/Moscow

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Dmitry Belyaev (University of Oxford)

Advanced lecture course "Geometry of smooth Gaussian fields"

Dmitry Belyaev ( University of Oxford)

The goal of this course is to give an introduction into the modern study of smooth Gaussian fields, their extremal values and geometry of their level sets. We will start with the general theory of Gaussian fields in $R^n$ and on other metric spaces. We will discuss their regularity and other analytic properties paying special attention to the case of stationary fields. After that, we will look at the property of level and excursion sets. First, we will prove Kac-Rice formulas and use them to compute the expected number of roots for Gaussian processes and the expected volume of level sets in higher dimensions. We will also cover other applications of these formulas. After that, we will turn our attention to non-local quantities such as the number of connected components of level sets. In particular, we will study recent break-through results of Nazarov and Sodin as well as the recent progress in understanding the large scale behaviour of planar level sets and their conjectured connection with percolation theory.

Additional pictures

• Bogomolny-Schmit.pdf
• convergence.pdf
• Fields_examples.pdf
• phase-transition.pdf

Videos

• Lecture1.mp4
• Lecture 2.mp4
• Lecture 3.mp4
• Lecture 4.mp4
• Lecture5.mp4

Whiteboard copy

• Lecture 1.pdf
• Lecture 2.pdf
• Lecture 3.pdf
• Lecture 4.pdf
• Lecture 5.pdf

Monday, 16 November

13:40 → 15:15 Lecture 1

Speaker: Dmitry Belyaev (University of Oxford)

In this lecture, I have explained how to construct a reproducing kernel Hilbert space (RKHS) associated with a smooth Gaussian field and proved a version of Kolmogorov's theorem which states that if the covariance kernel is Holder continuous, then the field is a.s. Holder continuous (with smaller exponent).

15:15 → 16:15 Consultation 1

Speaker: Dmitry Belyaev (University of Oxford)

Tuesday, 17 November

13:40 → 15:15 Lecture 2

Speaker: Dmitry Belyaev (University of Oxford)

We started with the Borell-TIS inequality and how jointly with the quantitative Kolmogorov theorem it allows controlling `bad' events when the field is much large than suggested by its point-wise variance. after that, we discussed Bulinskaya lemma which, in particular, was used to justify non-degeneracy assumptions behind Kac-Rice formula. We have sketched the proof of the one-dimensional Kac-Rice formula and have seen how it can be used to compute the expected number of real roots of a random polynomial. Finally, we looked at stationary fields. Using Bochner's theorem we can describe stationary fields in terms of their spectral measure. In particular, the field can be obtained as the Fourier transform of the white noise with respect to the spectral measure. 

15:15 → 16:15 Consultation 2

Speaker: Dmitry Belyaev (University of Oxford)

Wednesday, 18 November

13:40 → 15:15 Lecture 3

Speaker: Dmitry Belyaev (University of Oxford)

The convergence of the Random Spherical Harmonic to the Random Plane Wave and the convergence of the Kostlan ensemble to the Bargmann-Fock field. Theorems of Nazarov and Sodin about the existence of the density of nodal domains for a wide class of stationary fields. Bogomolny-Schmit conjecture about the connection between the critical bond percolation in Z^2 and nodal domains of the random plane wave.

15:15 → 16:15 Consultation 3

Speaker: Dmitry Belyaev (University of Oxford)

Thursday, 19 November

13:40 → 15:15 Lecture 4

Speaker: Dmitry Belyaev (University of Oxford)

Russo-Seymour-Welsh estimates for Gaussian fields. Results of Beffara-Gayet and B-Muirhead-Wigman on RSW for fast decaying fields and for the Kostlan ensemble on the sphere. All these results use quasi-independence results based on discretization ideas. 

15:15 → 16:15 Consultation 4

Speaker: Dmitry Belyaev (University of Oxford)

Friday, 20 November

13:40 → 15:15 Lecture 5

Speaker: Dmitry Belyaev (University of Oxford)

15:15 → 16:15 Consultation 5

Speaker: Dmitry Belyaev (University of Oxford)

Monday, 23 November

12:00 → 17:00 Consultation 6

Speaker: Dmitry Belyaev (University of Oxford)