TABLE OF CONTENTS The partition function or generating functional in field theory in the path integral approach involves the evaluation of multiple integrals over the local field variables where the dimension of the multiple integrals approaches infinity. In lattice gauge theory the dimension of the multiple integral over the link variables is finite, but very large. In many problems in statistical mechanics, for example the Ising model, one is faced with the evaluation of multidimensional sums over dynamic variables where the dimension of the sum is again typically very large. [1]
The goal of this section is to describe the numerical methods commonly used to accurately evaluate many-dimensional multiple integrals (or sums). This section is based on [Me53, Ne88, Du89]. Much of this material is taken from [Du89]. We start with a few preliminaries some statistics.
[1]Consider some numbers: In an LGT calculation in d dimensions with N sites along one axis, the number of links is dN d . For a lattice of size 16 4 in four dimensions the number of integrations over links in the multiple integrals=4 16 4=262, 144; this must still be multiplied by the number of internal link variables. For the Ising model, the number of spin configurations is 
. For a 64 64 lattice in 2 dimensions the number of spin configurations, which is the number of terms in the partition function sum, is 2 64 64=2 4096 ?10 1233!
Flip a coin N times.
