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HU Credits:
1
Degree/Cycle:
2nd degree (Master)
Responsible Department:
Mathematics
Semester:
1st Semester
Teaching Languages:
English
Campus:
E. Safra
Course/Module Coordinator:
Prof Maurice Duits
Coordinator Office Hours:
By Appointment
Teaching Staff:
Prof Jonathan Breuer
Course/Module description:
This course is given in English.
The aim of the course will be to give an overview of the large activity of
the past two decades on the theory of determinantal processes and their
appearance in various important models of probability and statistical me-
chanics. Determintantal point processes are random point processes where
the random points have a particular repulsive interaction. Even though we
break the independence that is characteristic to the classical and more well-
known Poissonian processes, there is still a beautiful mathematical structure
that allows one to study the quantities of interest. At the same time, many
important models from probability and statistical mechanics lead to de-
terminantal point processes. Because of their relevance and mathematical
elegance, they have been studied with great intensity in the past 20 years.
The development of the techniques for sturdying determinantal point pro-
cesses were crucial in proving long standing conjectures on, for example, the
longest increasing subsequence of a random permutation and universality
for eigenvalues of random matrices, and played an important role in study-
ing the asymptotic behavior of random tilings of large planar domains and
random growth processes. Applications of determinantal point processes are
numerous and go as far as wireless communication and machine learning.
The course will first deal with the foundations of determinantal point
processes, but quickly turn to the study of several important models from
mathematics and statistical physics. These will include, eigenvalues of ran-
dom Hermitian matrices, longest increasing subsequence, the Schur process
and random tilings of planar domains. Some of the highlights of the course
will be
-- Sine universality
-- Tracy-Widom distribution
-- Central limit Theorems for linear statistics
-- arctic circle for the domino tilings of the Aztec diamond
-- limit shapes and the Gaussian free eld for random tilings
A good knowledge of probability and analysis is recommended. Some of the
technical aspects of the course will require knowledge of complex analysis.
Course/Module aims:
After the course, the student should have a developed a broad overview of
how determinantal processes appear in various models of statistical physics
and mathematics. The student should also have developed sufficient compe-
tences to study technical papers in the field and even be sufficiently prepared
to start a research project in the area.
Learning outcomes - On successful completion of this module, students should be able to:
On successful completion of this module, the student should have a developed a broad overview of
how determinantal processes appear in various models of statistical physics
and mathematics. The student should also have developed sufficient compe-
tences to study technical papers in the field and even be sufficiently prepared
to start a research project in the area.
Attendance requirements(%):
none
Teaching arrangement and method of instruction:
lectures during the first two weeks of February
Course/Module Content:
The course will first deal with the foundations of determinantal point
processes, but quickly turn to the study of several important models from
mathematics and statistical physics. These will include, eigenvalues of ran-
dom Hermitian matrices, longest increasing subsequence, the Schur process
and random tilings of planar domains. Some of the highlights of the course
will be
Sine universality
Tracy-Widom distribution
Central limit Theorems for linear statistics
arctic circle for the domino tilings of the Aztec diamond
limit shapes and the Gaussian free field for random tilings
Required Reading:
none
Additional Reading Material:
Course/Module evaluation:
End of year written/oral examination 0 %
Presentation 0 %
Participation in Tutorials 0 %
Project work 100 %
Assignments 0 %
Reports 0 %
Research project 0 %
Quizzes 0 %
Other 0 %
Additional information:
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