This volume of proceedings stems from a school that was part of the programme Random Matrix Approaches in Number Theory, which ran at the Isaac Newton Institute for Mathematical Sciences, Cambridge, from 26 January until 16 July 2004. The purpose of these proceedings is twofold. Firstly, the impressive recent progress in analytic number theory brought about by the introduction of random matrix techniques has created a rapidly developing area of research. As a consequence there is not as yet a textbook on the subject. This volume is intended to fill this gap. There are, of course, well-established texts in both random matrix theory and analytic number theory, but very few of them treat in any length or detail these new applications of random matrix theory. Secondly, this new branch of mathematics is intrinsically multidisciplinary; teaching young researchers in random matrix theory, mathematical physics and number theory mathematical techniques that are not a natural part of their education is essential to introduce a new generation of scientists to this important and rapidly developing field. In writing their contributions to the proceedings, the lecturers kept in mind the diverse backgrounds of the audience to whom this volume is addressed.

The material in the volume includes the basic techniques of random matrix theory and number theory needed to understand the most important achievements in the subject; it also gives a comprehensive survey of recent results where random matrix theory has played a major role in advancing our understanding of open problems in number theory. We hope that the choice of topics will be useful to both the advanced graduate student and to the established researcher.

Abstract

What is a Mean Value Theorem?

By a mean value theorem we mean an estimate for the average of a function. When F(s) has a convergent Dirichlet series expansion in some half–plane Re s > σ0 of the complex plane, we typically take the average over a vertical segment:

The path of integration here need not lie in this half–plane. For example, we would like to know the size of the integrals

for σ ≥ ½ and k a positive integer. Here F(s) = ζ(s)k and its Dirichlet series converges only for σ > 1.

There are many variations. For example, one can consider a discrete mean value

where the points σr + itr lie in ℂ.

Having in mind the discussion on metastability, we need a brief introduction to statistical mechanics. This will be done in Section 3.3. Considerations on the connection between large deviation theory and the statistical description of (equilibrium) thermodynamical systems appear naturally and are the content of Section 3.4, focusing on basic aspects rather than generality, with references to more advanced literature.

Apart from such motivations, the extension of the results of Chapter 1 beyond independent variables is natural in many contexts. We discuss the Gärtner–Ellis method briefly and apply it to finite Markov chains, as the simplest situation to start with. While discussing large deviations for (equilibrium) statistical mechanics models, it is important to stress the role of subadditivity and convexity, already illustrated in Section 1.5, where the Ruelle–Lanford method was considered in the context of i.i.d. variables. Some basic results are taken from Section 3 of Pfister's lecture notes [243], simplified for our situation.

Large deviations for dependent variables. Gärtner–Ellis theorem

Examining the proof of the Cramér theorem in Section 1.4, we are naturally led to allow a moderate dependence among the Xi variables. This extension is due to Gärtner [132] and Ellis [108].

Introduction

In this chapter we discuss metastability and nucleation for several short range lattice spin systems at low temperature. We already analysed the case of the Curie–Weiss model in Chapter 4. Due to the fact that the intermolecular interaction does not decay at infinity, the Curie–Weiss model exhibits a mean field behaviour without any spatial structure; the configurations are well described, especially for large volumes, by the values of a unique macroscopic order parameter, i.e. the magnetization, so that the configuration space becomes one-dimensional. In contrast, for short range stochastic Ising models the geometrical aspects are particularly relevant. In this last case the configuration space can be viewed as a space of families of contours and it tends to be infinite dimensional in the thermodynamic limit. Indeed it is well known that the description of pure coexisting equilibrium phases at low temperature is naturally given in terms of a gas of contours (see Chapter 3, proof of Proposition 3.30, and [280]). It is clear that in the analysis of dynamical phenomena taking place in large finite systems, the geometrical description in terms of contours will also play a relevant role.

The central question in the description of the decay from a metastable to a stable phase for short range stochastic Ising models is, in addition to determination of the typical escape time, the characterization of the typical ‘nucleation pattern’ namely the typical sequences, in shape and size, of droplets along which nucleation of the stable phase takes place.

Introduction

In the analysis of a system with a large number of interacting components (at a microscopic level) it is of clear importance to find out about its collective, or macroscopic, behaviour. This is quite an old problem, going back to the origins of statistical mechanics, in the search for a mathematical characterization of ‘equilibrium states’ in thermodynamical systems. Though the problem is old, and the foundations of equilibrium statistical mechanics have been settled, the general question remains of interest, especially in the set-up of non-equilibrium systems. We could then take as the object of study a (non-stationary) time evolution with a large number (n) of components, where the initial condition and/or the dynamics present some randomness. One example of such a collective description is the so-called hydrodynamic limit. Passing by a space-time scale change (micro → macro) it allows, through a limiting procedure, the derivation of a reduced description in terms of macroscopic variables, such as density and temperature. Other limits, besides the hydrodynamic, may also appear in different situations, giving rise to macroscopic equations.

In all such cases the macroscopic equation indicates the typical behaviour in a limiting situation (n → +∞, and proper rescaling). Thus, it is essential to know something about:

1. rates of convergence, i.e. how are the fluctuations of the macroscopic random fields (for example, the empirical density) around the prescribed value given by the macroscopic equation?

2. how to estimate the chance of observing something quite different than what is prescribed by the macroscopic equation.

1. …