Department Colloquium
Department of Mathematics
Princeton University
February 28th, 2001
The longest paths in random matrices : connection to the
eigenvalues of random matrices
In the classical central limit theorem, the sum of i.i.d.
random variables has the same limiting fluctuation : the Gaussian
distribution. Now we make an M by N matrix
with entries of i.i.d. random variables, and take a maximum of sums of
entries along a class of up/right paths. Then we ask the question of
the limiting fluctuation as M and/or N tend to
infinity. This type of problem arises in for example, queueing theory and
interacting particle systems. There are a few examples for which one can
compute the limiting distribution directly, and
it turned out that the role played by the Gaussian distribution in the
classical central limit theorem is now played by the largest eigenvalue
of a random Hermitian matrix taken from the Gaussian
unitary ensemble. We will discuss topics centered around this relation
between the longest path in a random matrix and the largest eigenvalue
of a random Hermitian matrix.