in probability.

The first lecture discusses simple random walk in one dimension

and is anything but contemporary. It leads to a derivation of Stir-

ling's formula. The second lecture discusses random walk in several

dimensions and introduces the notion of power laws. Standard results

about probability of return to the origin and the intersection exponent

are discussed. The latter is a simply stated exponent whose value is

not known rigorously today, and it is a natural exponent to study by

simulation. The third lecture discusses the self-avoiding walk, which

is a very good example of a simply stated mathematical problem for

which most of the interesting questions are still open problems. The

fourth lecture considers the continuous limit of random walk, Brown-

ian motion. This topic was included to help those students who were

involved in simulations related to finance.

The next two lectures consider the problem of shuffling a deck of

cards. Lecture 5 discusses the general idea of random permutations

and introduces the notion of a random walk on a symmetric group.

The case of random riffle shuffles and the time (number of shuffles)

needed to get close to the uniform distribution was analyzed in a paper

of Bayer and Diaconis [BD], and Mann's paper [M] is an exposition

of this result. We give a short discussion of this result, although we do

not give all the details of the proof. This topic leads naturally to the

discussion of Markov chains and rates of convergence to equilibrium.

Lecture 7 is a standard introduction to Markov chains; it outlines a

proof of convergence to equilibrium that emphasizes the importance

of the size of the second eigenvalue for understanding the rate of

convergence. Lecture 8 disucsses a recent important technique to

sample from complicated distributions, Markov Chain Monte Carlo.

Lecture 9 discusses a very beautiful relationship between random

walks and electrical networks (see [DS] for a nice exposition of this

area). The basic ideas in this section are used in more sophisticated

probability; this is basically the discrete version of Dirichlet forms.

The work on electrical networks leads to the final lecture on uniform

spanning trees. We discuss one result that relates three initially quite