The random-field Ising model has been one of the big success stories of mathematical physics and deserves an entire chapter. It will give occasion to learn about many of the more powerful techniques available for the analysis of random systems. The central question heatedly discussed in the 1980s in the physics community was whether the RFIM would showspontaneous magnetization at lowtemperatures and weak disorder in dimension three, or not. There were conflicting theoretical arguments, and even conflicting interpretations of experiments. Disordered systems, more than others, tend to elude common intuition. The problem was solved at the end of the decade in two rigorous papers by Bricmont and Kupiainen (who proved the existence of a phase transition in d ≥ 3 for small ∈) and Aizenman and Wehr (who showed the uniqueness of the Gibbs state in d = 2 for all temperatures).

The Imry–Ma argument

The earliest attempt to address the question of the phase transition in the RFIM goes back to Imry and Ma in 1975. They tried to extend the beautiful and simple Peierls argument to a situation with symmetry breaking randomness. Let us recall that the Peierls argument in its essence relies on the observation that deforming one ground-state, +1, in the interior of a contour γ to another ground-state, −1, costs a surface energy 2|γ|, while, by symmetry, the bulk energies of the two ground-states are the same. Since the number of contours of a given length L is only of order CL, the Boltzmann factors, e–2βL, suppress such deformations sufficiently to make their existence unlikely if β is large enough.

The Laplacian acting on functions of finitely many variables appeared in the works of Pierre Laplace (1749–1827) in 1782. After nearly a century and a half, the infinite-dimensional Laplacian was defined. In 1922 Paul Lévy (1886–1971) introduced the Laplacian for functions defined on infinite-dimensional spaces.

The infinite-dimensional analysis inspired by the book of Lévy Leçons d'analyse fonctionnelleattracted the attention of many mathematicians. This attention was stimulated by the very interesting properties of the Lévy Laplacian (which often do not have finite-dimensional analogues) and its various applications.

In a work (published posthumously in 1919) Gâteaux gave the definition of the mean value of the functional over a Hilbert sphere, obtained the formula for computation of the mean value for the integral functionals and formulated and solved (without explicit definition of the Laplacian) the Dirichlet problem for a sphere in a Hilbert space of functions. In this work he called harmonic those functionals which coincide with their mean values.

In a note written in 1919 , which complements the work of Gâteaux, Lévy gave the explicit definition of the Laplacian and described some of its characteristic properties for the functions defined on a Hilbert function space.

In 1922, in his bookand in another publicationLévy gave the definition of the Laplacian for functions defined on infinite-dimensional spaces and described its specific features.

Martingales are a key tool of modern probability theory, in particular, when it comes to a.e. convergence assertions and related limit theorems. The origins of martingale techniques can be traced back to analysis papers by Kac, Marcinkiewicz, Paley, Steinhaus, Wiener and Zygmund from the early 1930s on independent (or orthogonal) functions and the convergence of certain series of functions, see e.g. the paper by Marcinkiewicz and Zygmund which contains many references. The theory of martingales as we know it now goes back to Doob and most of the material of this and the following chapter can be found in his seminal monograph from 1953.

We want to understand martingales as an analysis tool which will be useful for the study of Lp- and almost everywhere convergence and, in particular, for the further development of measure and integration theory. Our presentation differs somewhat from the standard way to introduce martingales – conditional expectations will be defined later in Chapter 22 – but the results and their proofs are pretty much the usual ones. The only difference is that we develop the theory for σ-finite measure spaces rather than just for probability spaces. Those readers who are familiar with martingales and the language of conditional expectations we ask for patience until Chapter 23, in particular Theorem 23.9, when we catch up with these notions.