Level-set percolation on the Discrete Gaussian Free Field (DGFF) turned out to be a hot topic within mathematical physics over the last couple of years. In particular, the DGFF on Z^d , with homogeneously weighted nearest-neighbour interactions, i.e. all conductances equal to 1, has been studied in detail. These models can be simulated with great efficiency. In this research, we abandon the homogeneity requirement and look at three-dimensional DGFFs with arbitrary conductances. Our goal is to find a quick and reliable method to simulate such DGFFs on a finite lattice. Since this is, in essence, a high-dimensional Gaussian sampling problem, we investigated this problem using the Conjugate Gradients (CG) linear solver as a Gaussian sampler. To see how it performed, we compared our implementation of the CG sampler with known methods for DGFFs in the unit conductance case. Finally, as a showcase of our implementation, we studied level-set percolation on a DGFF with a simple checkerboard conductance pattern. Our main conclusion is that the CG algorithm is very suitable for simulating Discrete Gaussian Free Fields. Since it does not make any assumptions on the conductances, it can be used to generate DGFFs with arbitrary conductances. However, there are still a number of issues with our implementation. The biggest one is concerning the stopping tolerance of the CG sampler. Once the tolerance is set smaller than some lattice size-dependent threshold, the percolative behaviour of the resulting sample changes drastically. We have not been able to explain this. Moreover, we would recommend making the implementation usable for parallel computing. We have been limited to relatively small lattice sizes during this project. Consequently, the use of certain finite-size scaling arguments when analysing level-set percolation might not always have been as justified. Finally, based on our study of the DGFF on a lattice with checkerboard conductances a and b (a < b), we conjectured that, in its percolative behaviour, this DGFF resembles a DGFF defined on a lattice with constant conductance c, where c is a weighted average of a and b. The weight of a is expected to be larger than the weight of b.

This thesis developed a computer powered simulation study of the divisible sandpile model. It introduces a constant to a widely used formula to generate sandpiles. This constant can be used to study the convergence characteristics of sandpiles. In this thesis it is shown that the introduced factor increases the speed of convergence while it no longer stabilizes in the all-1-configuration. While one should be careful with using this constant as it does not work for every distribution, it does speed up the stabilization significantly. There is also a positive correlation found between the constant and the amount of nodes that remain lower than 1 in the stable state.

For modelling microstructures of materials the Voronoi diagram is one of the most commonly used models. In this thesis we study a generalization of Voronoi diagrams known as the Laguerre-Voronoi diagram. In particular, we consider the stereological problem of estimating the 3D cell volume distribution of such a diagram from one of its 2D planar sections. This problem is not in general solvable for all Laguerre-Voronoi diagrams. We consider a specific class of Laguerre-Voronoi diagrams generated using an algorithm which allows to control the volume distribution of its cells. Inspired by related stereological problems, an estimator for the distribution of areas observed in a planar section is proposed. This estimator can be used for estimating the cell volume distribution from a sample of observed areas in a planar section of a 3D Laguerre-Voronoi diagram. Given that the problem is motivated by a materials science application we consider Laguerre-Voronoi diagrams with a lognormal cell volume distribution, which is commonly used in this field. The simulations show that the proposed method works well in the sense that on average the estimated parameters of the lognormal cell volume distribution are close to the actual parameter values. While the focus is on the lognormal distribution, generalization of the estimator considering other parametric distributions is briefly discussed. The thesis is concluded with a case study: the 3D grain volume distribution of a real steel microstructure is estimated from observed areas obtained from 2D image data.

The random graph is a mathematical model simulating common daily cases, such as ranking and social networks. Generally, the connection between different users in the network is established through preference, and this phenomenon leads to a power-law behaviour of the degree sequence of the random graph. Other than studying this feature, the thesis also focuses on the decay of degree sequence itself and the height of the random graph models. Researches are conducted on four kinds of random graph models, including two fully studied modes as benchmarks, the Preferential Attachment Model (PAM) and Random Recursive Tree (RRT), and another common model, the Bianconi-Barabási Model (BBM). Previous researches show that the degree sequence of PAM follows a power-law with coefficient -3, and the degree sequence of RRT has an exponential tail in contrast. In BBM, each node is assigned a fitness value from a certain distribution, once the node is generated, and researches show that the degree sequence of BBM follows a power-law with coefficient depending on the fitness distribution but close to the one of PAM. Then, the thesis aims to find a random graph model with fitness interpolating between RRT and PAM. Thus, a new Recursive Fitness Model (RFM) is developed by using a recursive formula to generate fitness values instead of the fixed distribution in BBM. By simulation we conjecture that the degree sequences of RFMs still follow a power-law with coefficient close to that of PAM. The simulation further shows an interesting phenomenon: a special linear height increment of a particular RFM, the Plus-1 model, appears, in contrast to other RFM models. Finally, the thesis invents a new Preferential Attachment Inverse Model (PAMinv) by using the expectation of degree sequence of PAM. Degree sequence of PAMinv shows a different behaviour from PAM and RRT, and it could be the tool to interpolate RRT from PAM.