Random matrices
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Random matrices appear in many areas of science and engineering. They are used to model large networks, complex systems in physics, and also appear in signal processing, for example.
Our group focuses on the theoretical aspects of random matrices, and more specifically on their eigenvalues. This research is closely connected to probability theory and complex analysis, and many of the problems we study are motivated by questions from mathematical physics.
Non-Hermitian matrices
Eigenvalues of a Ginibre matrix of size 250 x 250
250 i.i.d. uniformly distributed random variables on the unit disk
The points on the left appear more evenly spaced on the unit disk than those on the right.
This suggests that the eigenvalues of Ginibre matrices are not independent but repel each other.
The following well-known result confirms this expectation:
The above double product implies that the density vanishes quadratically when two eigenvalues approach each other, which explains the repulsion mentioned above.
Our group studies several interesting aspects of the eigenvalues of Ginibre matrices, including: :
Counting statistics: determining the limiting distribution of the number of eigenvalues within a given subset of the unit disk.
Hole probabilities: determining the probability that a particular subset of the unit disk contains no eigenvalues.
Gap statistics: determining the limiting distribution of the smallest and largest gaps between neighboring eigenvalues.
Local statistics: determining the typical behavior of the eigenvalues on a local scale.
There exist many other models of non-Hermitian matrices that our group studies.
Examples of eigenvalues of such matrices are represented in the following figures:
Hermitian matrices
As for Ginibre matrices, the eigenvalues of GUE matrices are independent of the eigenvectors. Moreover, the following holds:
Histogram showing the eigenvalue density of a GUE matrix of size 2000 x 2000 (orange), together with Wigner's semi-circle law (blue).
The above is a macroscopic result. It is also interesting to study the local eigenvalue statistics.
The next figures compare 20 eigenvalues in the bulk of a large GUE matrix (middle) with independent random variables (left) and deterministic points (right).
20 i.i.d. uniform(0,1)
20 eigenvalues of a GUE matrix in the bulk (i.e. in the "middle" of the spectrum)
20 equispaced points between 0 and 1
As can be seen from the above pictures, the eigenvalues of GUE matrices seem to exhibit less randomness than independent random variables. In particular,
the smallest gaps appear smaller on the left,
the largest gaps appear larger on the left.
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