Finite-size effects in high-dimensional statistical mechanical systems: The Ising model with periodic boundary conditions

A fundamental observation in statistical mechanics is that the behavior of many systems simplifies considerably above the upper critical dimension: for example, the critical exponents characterizing a phase transition attain their mean-field values. However, there are important quantities, such as finite-size corrections and bulk quantities, on which this paradigm offers contradicting predictions. Renormalization group arguments have been suggested to account for this; however, they are phenomenological, unrigorous, and lack intuition. In contrast, the random-geometric approach offers a natural explanation for the existence of the upper critical dimension, and a simple resolution of the apparent contradiction.; This thesis is composed of two parts: the first contains overviews of the high-dimensional behavior of thermodynamical systems, and the regularization and renormalization procedures in constructive field theory, in order to expose the significance of the Ising model in addressing important issues in statistical mechanics and higher energy physics. This is followed by an intuitive introduction to the random-geometric approach, which has been successful in addressing some of these issues; we use it to motivate new results on the Ising model, regarding the effect of boundary conditions on the scaling limit. These results are consonant with recent findings in the theories of percolation and loop-erased random walks, which contribute to an emerging picture of multi-scale criticality.; The second part of the thesis consists of the rigorous study of the ferromagnetic Ising model on a box ΛL of linear size L above the upper critical dimension duc = 4, using algebraic and differential inequalities derived with the current and random-random-path representations. The differential inequalities are modified versions of results appearing in the work of previous authors, while the algebraic inequalities are first derived here, through coupling the system with periodic boundary conditions to the one in the infinite lattice. We also use the random-current representation to control the long-range fluctuations of the two-point function.; These methods are then used to confirm the basic ingredients of the intuitive picture described in the first part. The short-range behavior of the model is essentially unaffected by the choice of boundary conditions, and can be understood in terms of non-intersecting, independent simple random walks; this is reflected in the behavior of bulk quantities, at temperatures where the correlation length is small enough.