This thesis is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.

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Consider random sequential adsorption on a chequerboard lattice with arrivals at rate 1 on light squares and at rate λ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites black, and the remaining squares white. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability p; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs (λ, p), containing the pair (1,0.5), such that for (λ, p) lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates. We find conditions satisfied by a broad class of essentially planar percolation models such that for a model satisfying the conditions, the presence or absence of percolation is determined by what happens in a collection of finite boxes. This criterion applies to a (non-degenerate) Poisson Boolean model, to the random connection model for some sufficiently high p < 1, and to the model described above. We also find conditions that do not require rotation invariance which produce a comparable result; these conditions seem plausible, but finding non-trivial examples is a matter for further research.

The derivation of the Boltzmann equation from a particle model of a gas is currently a major area of research in mathematical physics. The standard approach to this problem is to study the BBGKY hierarchy, a system of equations that describe the distribution of the particles. A new method has recently been developed to tackle this problem by studying the probability of observing a specific history of events. We further develop this method to derive the linear Boltzmann equation in the Boltzmann-Grad scaling from two similar Rayleigh gas hard-sphere particle models. In both models the initial distribution of the particles is random and their evolution is deterministic. Validity is shown up to arbitrarily large times and with only moderate moment assumptions on the non-equilibrium initial data. The first model considers a Rayleigh gas whereby one tagged particle collides with a large number of background particles, which have no self interaction. The initial distribution of the background particles is assumed to be spatially homogeneous and at a collision between a background particle and the tagged particle only the tagged particle changes velocity. In the second model we make two changes: we allow the background particles to have a spatially non-homogeneous initial data and we assume that at collision both the tagged particle and background particle change velocity. The proof for each model follows the same general method, where we consider two evolution equations, the idealised and the empirical, on all possible collision histories. It is shown by a semigroup approach that there exists a solution to the idealised equation and that this solution is related to the solution of the linear Boltzmann equation. It is then shown that under the particle dynamics the distribution of collision histories solves the empirical equation. Convergence is shown by comparing the idealised and empirical equations.