Gaussian Fields: Geometry and Applications in Random Matrix Theory

June 10 afternoon to June 11th morning, 2019, at Queen Mary University of London. Thanks everybody for a wonderful workshop!

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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