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Point processes and random measures

External Lecturer: 
Alexander Bufetov
Course Type: 
PhD Course
Academic Year: 
20 h

The theory of point process is soon turning 500, yet very many questions remain unanswered.

The first part of the course will give a gentle introduction to point processes, with a special emphasis on determinantal point processes.

The second part will be devoted to corresponding random measures such as the Gaussian multiplicative chaos. We will start from the basics, no prerequisites assumed and will go on to topics of current research.

The study of point processes, that is, random subsets of a Polish space, goes back at least to the 1662 work of John Graunt on mortality in London. Matrices whose entries are given by chance were studied by Ronald Fisher in 1915 and John Wishart in 1928 and used by Freeman Dyson who in 1962 observed  that “the statistical theory (…) will describe the degree of irregularity (…) expected to occur in any nucleus”.  

The Weyl character formula implies that the correlation functions for the eigenvalues  of a Haar-random unitary matrix have determinantal form --- the Ginibre-Mehta theorem, --- and in 1973 Odile Macchì started the systematic study of point processes whose correlation functions are given by determinants.

This level of abstraction has proved to be very fruitful: on the one hand, examples of determinantal point processes arise in diverse areas such as asymptotic combinatorics (Burton-Pemantle, Benjamini-Lyons-Peres-Schramm, Baik-Deift-Johansson, Borodin-Okounkov-Olshanski), representation theory of infinite-dimensional groups (Olshanski, Borodin-Olshanski), random series (Hough- Krishnapur-Peres-Virág) and, of course, random matrices;  on the other hand, the general theory of determinantal point processes includes limit theorems (Soshnikov),  a characterization of Palm measures (Shirai-Takahashi), the Kolmogorov as well as the Bernoulli property (Lyons, Lyons-Steif), and rigidity (Ghosh, Ghosh-Peres). 

In this course, the correlation kernels of our determinantal point processes will be assumed to induce orthogonal projections: for example, the sine-kernel of Dyson induces the projection onto the Paley-Wiener space of functions whose Fourier transform is supported on the unit interval, while the Bessel kernel of Tracy and Widom induces the orthogonal projection onto the subspace of square-integrable functions whose Hankel transform is supported on the unit interval.  

One of the key questions discussed in the course is the following. What is the relation between the point process and the  Hilbert space that governs it?

Extending earlier work of Lyons and Ghosh, in joint work with Qiu and Shamov  it is proved that almost every realization of a determinantal point process is a uniqueness set for the underlying Hilbert space. For the sine-process, almost every realization with one particle removed is a complete and minimal set for the Paley-Wiener space, whereas if two particles are removed, then one obtains a zero set for the Paley-Wiener space. Quasi-invariance of the sine-process under compactly supported diffeomorphisms of the line plays a key rôle.

We will then proceed to the problem of sampling functions from their restrictions onto a realization of a determinantal point process.

The 1933 Kotelnikov theorem samples a Paley-Wiener function from its restriction onto the integers. How to reconstruct a Paley-Wiener function from a realization of the sine-process?

In joint work with Borichev and Klimenko, it is proved that if a Paley-Wiener function decays at infinity as a sufficiently high negative power of the distance to the origin, then the  Lagrange interpolation formula yields the desired reconstruction. Similar results are also obtained for the Airy kernel, the Bessel kernel  and the Ginibre kernel  of orthogonal projection onto the Fock space. 

In joint work with Qiu, the Patterson-Sullivan construction is used to interpolate Bergman functions from a realization of the determinantal point process with the Bessel kernel, in other words, by the Peres-Virág theorem, the zero set of a random series with independent complex Gaussian entries. The invariance of the zero set under the isometries of the Lobachevsky plane plays a key rôle. 

The second part of the course will provide an introduction to the Gaussian Multiplicative Chaos, the multiplicative analogue of Brownian motion introduced by Jean-Pierre Kahane.

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