Gaussian Fields: Geometry and Applications in Random Matrix Theory

June 10 afternoon to June 11th morning, 2019, at Queen Mary University of London, Queen's Building, Mile End campus (building 19 on this campus map, directions to campus here). All talks are in LG1 in the basement (Note the change of room!). All breaks are in SCR bar on the second floor of Queen's Building.

Organizers: Anna Maltsev (annavmaltsev@gmail.com), Stephen Muirhead (s.muirhead@qmul.ac.uk)


Travel funding is still available for students and early career researchers. Please send an inquiry and a cv to Anna Maltsev at annavmaltsev@gmail.com.

There is no registration fee. Please confirm your attendace by filling the registration form.
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Program:
June 10
1pm - 1:55pm Yan Fyodorov On energy landscapes of elastic manifolds in random potentials
2 pm - 2:55pm Anna Maltsev Covariance formulas for fluctuations of linear spectral statistics for Wigner matrices with few moments

3pm - 3:30pm Coffee break

3:30pm - 4:25pm Jerry Buckley Fluctuations in the zero set of stationary Gaussian processes
4:30 - 5:25 pm Anne Estrade On Berry's dislocation lines in 3D framework

5:30 - 7pm Reception
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June 11
9:30 - 10:25am Stephen Muirhead A covariance formula for topological events of smooth Gaussian fields
10:30 - 11:00am Michael McAuley The variance of the number of excursion sets of planar Gaussian fields

11:00 - 12:00am Coffee and sandwiches

12:00 - 12:30pm Mihail Poplavskyi On the persistence probability for the sech-correlated Gaussian Stationary Process
12:35 - 13:30pm Paul Bourgade The characteristic polynomial of Ginibre matrices
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Abstracts:

Paul Bourgade. The characteristic polynomial of Ginibre matrices.
Abstract. This is joint work with Guillaume Dubach and Lisa Hurting. I will explain the calculation of the joint moments of the characteristic polynomial, for non-Hermitian matrices from the complex Ginibre ensemble. As a consequence, this characteristic polynomial converges to the two dimensional Gaussian multiplicative chaos.

Jerry Buckley. Fluctuations in the zero set of stationary Gaussian processes.
Abstract: The mean number of zeroes of a stationary Gaussian process is given by the famous Kac-Rice formula. A formula in a similar spirit, due to Cramer-Leadbetter, computes the variance exactly. Unfortunately this expression is not very accessible, and it is difficult to get a good estimate for the size of the variance. We will propose an approximate formula for a general process, that allows one to compute the asymptotic growth of the variance. In particular we show that the variance always grows at least linearly for a non-trivial process. Work in progress with Eran Assaf and Naomi Feldheim.

Anne Estrade. On Berry's dislocation lines in 3D framework.
Abstract: We study the length of dislocation lines that are given as nodal lines of a complex Gaussian random wave indexed by R^3. We consider random waves that are eigenfunction of the 3D Euclidean Laplacian, isotropically distributed or not. We are particularly interested in the expectation and the variance of the length and their behavior as the associated eigenvalue goes to infinity. We compare the results with the similar situation in 2D that has been extensively studied since Berry's paper in 2002. This is a work in progress with Federico Dalmao and Jose Leon (Universidad de la Republica, Uruguay).

Anna Maltsev. Covariance formulas for fluctuations of linear spectral statistics for Wigner matrices with few moments.
Abstract: We examine covariance of the fluctuations of the linear spectral statistics for Wigner matrices with entries that have a finite variance but no finite 4th moment. We obtain a closed form expression for the covariance (previously expressed as a double contour integral) and we compute the relevant integral kernel. We furthermore place it in context by a comparison with a similar covariance arising in the light tailed case, i.e. when the entries have more than 4 moments. This is joint work with Asad Lodhia.

Michael McAuley The variance of the number of excursion sets of planar Gaussian fields.
Abstract: Nazarov and Sodin have shown that the expected number of connected components of the nodal set of a smooth Gaussian field in a large domain is proportional to the volume of the domain. In this talk I focus on planar Gaussian fields and discuss new lower bounds on the variance of the number of excursion/level set components at arbitrary levels. In particular I will give bounds at certain levels for two special cases of interest: the Random Plane Wave and the Bargmann-Fock field. This talk is based on joint work with Dmitry Belyaev and Stephen Muirhead.

Stephen Muirhead A covariance formula for topological events of smooth Gaussian fields.
Abstract: A topological event for a smooth random field is an event that depends only on the topology of a level set of the field. Examples include events that depend on the number of level set components, and the event that a level set crosses a rectangle from left to right. We derive an exact formula for the covariance between any topological events of a smooth Gaussian field. Simple applications include strong mixing bounds for topological events, lower concentration for topological counts, and an alternative proof of the FKG inequality for positive-correlated Gaussian fields. The covariance formula is based on a classic interpolation argument of Piterbarg. Joint work with Dmitry Beliaev and Alejandro Rivera.


Funded by LMS and QMUL.

Conference poster