  The theory of large deviations deals with the probabilities of rare events
(or fluctuations) that are exponentially small as a function of some parameter,
e.g., the number of random components of a system, the time over which a
stochastic system is observed, the amplitude of the noise perturbing a
dynamical system or the temperature of a chemical reaction. The theory has
applications in many different scientific fields, ranging from queuing theory
to statistics and from finance to engineering. It is also increasingly used in
statistical physics for studying both equilibrium and nonequilibrium systems.
In this context, deep analogies can be made between familiar concepts of
statistical physics, such as the entropy and the free energy, and concepts of
large deviation theory having more technical names, such as the rate function
and the scaled cumulant generating function. The first part of these notes
introduces the basic elements of large deviation theory at a level appropriate
for advanced undergraduate and graduate students in physics, engineering,
chemistry, and mathematics. The focus there is on the simple but powerful ideas
behind large deviation theory, stated in non-technical terms, and on the
application of these ideas in simple stochastic processes, such as sums of
independent and identically distributed random variables and Markov processes.
Some physical applications of these processes are covered in exercises
contained at the end of each section. In the second part, the problem of
numerically evaluating large deviation probabilities is treated at a basic
level. The fundamental idea of importance sampling is introduced there together
with its sister idea, the exponential change of measure. Other numerical
methods based on sample means and generating functions, with applications to
Markov processes, are also covered.
