CHAPTER 4
THE MONTE CARLO APPROACH
This chapter presents the second theme of the thesis, namely the numerically
implemented Monte Carlo approach to studying the Tr(o3) field theory in two dimensions.
We will see why, in this case, such a stochastic approach is preferred over a deterministic
numerical investigation. The mathematical framework for Markov chains is well
established and is discussed in detail in basic textbooks on the subject [18].
4.1 Introduction to Monte Carlo Techniques
Since the advent of easy, ubiquitous access to powerful computers, numerical methods
have come to be widely used in physics. Quantum field theory is no exception to this.
Numerical methods can, in principle, be categorized as deterministic or stochastic. Both
types yield approximate answers but in the case of deterministic methods the error can
be traced back to the finite-precision representation of real numbers used in computers.
As the name implies, deterministic methods operate in a predefined manner. Stochastic
methods, on the other hand, rely heavily on statistics and errors originate not only from
floating point representation of numbers, but also from the statistical interpretation of the
results, since random numbers are used as inputs of the simulation. Monte Carlo methods
are examples of stochastic numerical methods and ],,. v are so widely used that the term
" Ionte Carlo" is sometimes taken to mean any stochastic numerical method.