Abstract:
Random walks are also called Markov chains. They have
applications in physics, statistics, biology, games, and in
other areas of mathematics.
In a random walk we move around in a network, step by step.
From any node in the network, we will move along one of the
arcs to an adjacent node; each arc has a pre-assigned chance
to be chosen.
When the network is finite, we organize the step
probabilities into a transition matrix. High powers of the
transition matrix usually converge to determine a stable
distribution. In fact, any finite network has a stable
distribution which can be found by solving a system of
equations coming from the transition matrix.
However, the concept of the stable distribution is not
limited only to finite networks. A random walk on an
infinite network which is "positive recurrent" will also
have a stable distribution. We will create some positive
recurrent random walks on infinite networks; and solve some
infinite systems of equations to find their stable
distributions.