Large deviations and metastability.

*(English)*Zbl 1075.60002
Encyclopedia of Mathematics and Its Applications 100. Cambridge: Cambridge University Press (ISBN 0-521-59163-5/hbk). xv, 512 p. (2004).

This book yields a wide overview of the current mathematical understanding of metastability phenomena arising in the context of Markov processes. Such phenomena were first described in the theoretical physics and chemistry literature, and many of the models discussed in the book are connected to statistical mechanics. This fact enables the authors to provide much physical intuition regarding the phenomena under study; however, their aim is to provide a mathematical treatment of metastability, and the proofs of the main theorems are either given in full detail or carefully outlined, with a precise reference to the relevant articles in the mathematical litterature.

The main mathematical tools used throughout the book are large deviations and Freidlin-Wentzell estimates, and the first three chapters provide a streamlined introduction to these tools. The first chapter presents large deviation bounds in the context of sums of i.i.d. random variables (Cramér-Chernoff and Sanov theorems), and the reader is introduced along the way to some of the essential principles arising in the computation of LD bounds (contraction principle, Varadhan’s lemma). The second chapter is an introduction to Itô diffusion processes, comprising the Itô formula as well as Girsanov’s theorem, followed by a presentation of basic Freidlin-Wentzell estimates for the large deviation properties of such diffusions in the small noise limit. The third chapter is again devoted to general large deviation estimates, this time in the context of certain dependent families of random variables: Markov chains are treated first using the Gärtner-Ellis theorem, and the authors then turn to the large deviation properties of certain Gibbs fields on the lattice \(Z^d\), after giving a brief introduction to equilibrium statistical mechanics.

Chapter 4 might be considered as bearing the greatest significance upon a first reading, since it provides a detailed and very accessible introduction to metastability; the phenomenon is considered both in the context of gas condensation and of ferromagnetism, and different approaches are presented successively, namely the van der Waals-Maxwell theory and the Curie-Weiss mean field theory, the Kac potential and Lebowitz-Penrose theory, the classical theory of nucleation, and finally the pathwise approach to metastability. Special attention is paid to this last approach, which may be summed up as follows: the stochastic process under consideration has a unique stationary measure and one aims at proving that, for suitable initial conditions, it behaves for a long time (i.e. on a suitable time scale \(R_N\nearrow \infty\), where \(N\gg1\) stands for the size of the system) as if it were described through another equilibrium measure (“thermalisation in the metastable state”); eventually, and unpredictably, the system undergoes a transition to its true equilibrium (“tunneling”). This approach is given an immediate illustration, first in the context of the Curie-Weiss mean field model of ferromagnetism, then for the supercritical contact process in one dimension, and the chapter ends with an evocation of some possible extensions, e.g. to random field Curie-Weiss models or contact processes in higher dimensions.

The remaining three chapters are devoted to several other applications of the pathwise approach to metastability: Chapter 5 deals with the metastability of certain diffusion processes considered in the small noise limit, using the basic ingredients of Chapter 2 together with a coupling method for diffusions that was first proposed by Lindvall and Rogers. Chapter 6 is a discrete pendant to Chapter 5 and deals with the metastability of certain reversible Markov chains. Finally, Chapter 7 is devoted more specifically to the metastable behaviour of certain lattice spin systems in the low temperature regime, building upon the results of Chapter 6; greater attention is paid to the Ising model of ferromagnetism, and an alternative approach to the metastability of certain spin systems is also briefly outlined, the references to recent work of Bovier and his collaborators being given in detail.

The main mathematical tools used throughout the book are large deviations and Freidlin-Wentzell estimates, and the first three chapters provide a streamlined introduction to these tools. The first chapter presents large deviation bounds in the context of sums of i.i.d. random variables (Cramér-Chernoff and Sanov theorems), and the reader is introduced along the way to some of the essential principles arising in the computation of LD bounds (contraction principle, Varadhan’s lemma). The second chapter is an introduction to Itô diffusion processes, comprising the Itô formula as well as Girsanov’s theorem, followed by a presentation of basic Freidlin-Wentzell estimates for the large deviation properties of such diffusions in the small noise limit. The third chapter is again devoted to general large deviation estimates, this time in the context of certain dependent families of random variables: Markov chains are treated first using the Gärtner-Ellis theorem, and the authors then turn to the large deviation properties of certain Gibbs fields on the lattice \(Z^d\), after giving a brief introduction to equilibrium statistical mechanics.

Chapter 4 might be considered as bearing the greatest significance upon a first reading, since it provides a detailed and very accessible introduction to metastability; the phenomenon is considered both in the context of gas condensation and of ferromagnetism, and different approaches are presented successively, namely the van der Waals-Maxwell theory and the Curie-Weiss mean field theory, the Kac potential and Lebowitz-Penrose theory, the classical theory of nucleation, and finally the pathwise approach to metastability. Special attention is paid to this last approach, which may be summed up as follows: the stochastic process under consideration has a unique stationary measure and one aims at proving that, for suitable initial conditions, it behaves for a long time (i.e. on a suitable time scale \(R_N\nearrow \infty\), where \(N\gg1\) stands for the size of the system) as if it were described through another equilibrium measure (“thermalisation in the metastable state”); eventually, and unpredictably, the system undergoes a transition to its true equilibrium (“tunneling”). This approach is given an immediate illustration, first in the context of the Curie-Weiss mean field model of ferromagnetism, then for the supercritical contact process in one dimension, and the chapter ends with an evocation of some possible extensions, e.g. to random field Curie-Weiss models or contact processes in higher dimensions.

The remaining three chapters are devoted to several other applications of the pathwise approach to metastability: Chapter 5 deals with the metastability of certain diffusion processes considered in the small noise limit, using the basic ingredients of Chapter 2 together with a coupling method for diffusions that was first proposed by Lindvall and Rogers. Chapter 6 is a discrete pendant to Chapter 5 and deals with the metastability of certain reversible Markov chains. Finally, Chapter 7 is devoted more specifically to the metastable behaviour of certain lattice spin systems in the low temperature regime, building upon the results of Chapter 6; greater attention is paid to the Ising model of ferromagnetism, and an alternative approach to the metastability of certain spin systems is also briefly outlined, the references to recent work of Bovier and his collaborators being given in detail.

Reviewer: Michel Sortais (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60F10 | Large deviations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |

82C22 | Interacting particle systems in time-dependent statistical mechanics |