This thesis contains a rigorous derivation of the path integral formulation of the Isingmodel with multiple original proofs. Besides that, thesis also contains various resultsof simulations of the 2D square lattice Ising Model with nearest-neighbour interactionsusing the Swendsen-Wang algorithm. Using finite size scaling to find critical exponentγ, we reported a value of γ = 1.748 ± 0.004. After calculating the relaxation times τ forvarious thermodynamic variables, we found the value for the dynamic critical exponentz to be in the range of z = 0.180 ± 0.004 and z = 0.282 ± 0.005.

The behaviour of the dilute Curie-Weiss model has been analysed for Gaussian, bounded and subGaussian couplings. This model is an abstraction of the Curie-Weiss model which is a model of a spin configuration where instead of a constant coupling between spins the spin coupling is a realisation of a random variable. To analyse this behaviour first the behaviour of the standard Curie-Weiss model is depicted. A notable part from this behaviour is the presence of a phase transition., indicating that this model can be used for a study of phase transitions. After having analysed the behaviour of the standard Curie-Weiss model theorems, stating the closeness between the standard Curie-Weiss model and the randomly dilute Curie-Weiss model for Gaussian, bounded and sub-Gaussian couplings, will be proven. Indicating that the models behave approximately the same way. These theorems will be in the form of a bound for the on randomly dilute Curie-Weiss model as an exponential multiplied by the standard Curie-Weiss model with the probability of this bound being a sub-Gaussian distribution.

Markov chains are used to describe random processes in discrete time, which have the property of being memoryless. This report covers Markov chains on a finite space that are homogeneous in time and mainly follows the structure of ''Markov Chains and Mixing Times''. Markov chains exhibit a strong connection with electric networks. We exploit such a connection and apply the laws of physics to answer several probabilistic questions about random walks, which are certain types of Markov chains. This connection is then called the electric network approach and is built on translating the random walk into an electric network by relating the transition probability to the so-called conductance. The electric network approach provides problem simplification tools, such as the Series/Parallel Law, and powerful inequalities, such as the Nash-Williams inequality. These tools are based on physics and are often more intuitive than their probabilistic counterpart.

We simulate a two-dimensional random walk that starts at the center of a square and ''escapes'' if it reaches the perimeter of the square before returning to the center. We then compare this escape probability to an upper bound, which results from using the Nash-Williams inequality. The sharpness of the upper bound depends on the choice of edge-cutsets. We find that choosing edge-cutsets with a minimal amount of edges gives a sharper upper bound, than choosing edge-cutsets that contain all the edges of the square. The relation between the upper bound and the escape probability seems to be independent of the size of the square. Furthermore, we provide proofs that are not explicit in ''Markov Chains and Mixing Times'' and ''Reversible Markov Chains and Random Walks on Graphs'' and add to the contents of ''Markov Chains and Mixing Times'' by studying random walks from a graph theory perspective.