Eigenvalues distributions of Toeplitz and Hankel matrices
A square matrix whose negative-sloping diagonals have constant
values is called Toeplitz matrix. In other words, the (i,j)
element of a Toeplitz matrix depends only on the difference i - j
of the row and column indices. Instead, the (i,j) entry of a
Hankel matrix depends only of the sum i + j of the indices. To
each Toeplitz or Hankel matrix it is possible to associate a
function called symbol of the matrix. It is known for Toeplitz
matrices that, for a certain class of singular symbols, the
eigenvalues of matrices of high dimension tend to align themselves
along the image of the symbol. Little is known about the
behaviour of the eigenvalues of Hankel or of sum of Toeplitz and
Hankel matrices. This project consists of investigating
numerically the distribution of the eigenvalues of Hankel matrices
and of combinations of Toeplitz and Hankel matrices. The amount
of work can be calibrated for both a 10cp and a 20cp project.
Roots of random polynomials
In 1943 Kac discovered that the expected number of real zeros of a
random polynomial of degree n, whose coefficients are real
independent identically distributed random variables, is
approximately (2/pi)*log n for large n. This project involves
investigating various aspects of the behaviour of the roots of
random polynomials. Its direction will be determined by the
interests of the student. For example, possible investigations
include: determining numerically the correction to Kac's formula;
studying the distribution and correlations of roots of random
polynomials whose coefficients are not independent random
variable; investigating the distribution of the zeros of
characteristic polynomials of matrices whose elements are real
random variable. The amount of work can be calibrated for both a
10cp project and a 20cp project.
Level 4 projects
Derivatives of characteristic polynomials of random unitary
matrices
The roots of the characteristic polynomials of random unitary
matrices are distributed on the unit circle. A theorem of Gauss
implies that the roots of their derivatives lie all inside the
unit circle. It is a natural question to ask how they are
distributed. Recent studies show that as the dimension of the
matrices increases, they tend to migrate toward the boundary of
the circle. This project involves studying this distribution, not
only for the group of unitary matrices but also for other
ensembles of random matrices. These studies are closely related
to the investigation of the zeros of the Riemann zeta function and
of other important functions in number theory known as
L-functions. 20cp project.
Entanglement in quantum spin chains and random matrix theory
Recent studies have shown that random matrix theory can give
insight into the understanding of phase transitions in quantum
spin chains. In particular it can be used to compute the
entanglement between two parts of the chain when the system is in
the ground state. This project involves investigating numerically
and analytically entanglement in various families of quantum spin
chains using these new techniques. 20cp project.
