The self-avoiding walk.

*(English)*Zbl 0780.60103
Probability and Its Applications. Boston, MA: Birkhäuser. xiv, 425 p. (1993).

A self-avoiding walk (SAW) on a lattice is a path which visits no site more than once. The SAW is a mathematical model which has important applications in statistical mechanics and in polymer science. In spite of the simple definition of SAW, the properties of typical SAW’s are difficult to analyse rigorously. For example, what is the distance between the endpoints of a typical SAW of length \(n\) on the square lattice? The answers to this and to other important questions remain unknown in a rigorous mathematical sense, although physicists and chemists have well developed but non-rigorous techniques to approach such questions. Mathematicians also have made much recent progress. The book under review is the first unified account of the rigorous theory. “Its goals are to give an account of the current mathematical understanding of the model, to indicate some of the applications of the concept in physics and in chemistry, and to give an introduction to some of the non-rigorous methods used in those fields.”

For simplicity, consider SAW’s on the \(d\)-dimensional hypercubic lattice \(\mathbb{Z}^ d\), where \(d\geq 2\). The two basic questions are

(a) how many SAW’s of length \(n\) exist,

(b) how far apart (on the average) are the endpoints of such a SAW?

In a famous paper, J. M. Hammersley and J. G. Morton [J. Roy. Stat. Soc., Ser. B 16, 23-38 (1954; Zbl 0055.369)] showed that the number \(c_ n\) of such walks satisfies \(c_{m+n}\leq c_ mc_ n\), whence by subadditivity the connective constant \(\mu=\lim_{n\to\infty}c_ n^{1/n}\) exists. The power series \(\chi(z)=\sum_ nc_ nz^ n\) has radius of convergence \(z_ c=\mu^{-1}\), and \(z_ c\) plays the role of the critical point of the model (akin to the critical temperature of a model of statistical mechanics). A principle target is to understand the behaviour of \(\chi\) and other related power series in the vicinity of the singularity \(z_ c\): what is the nature of this singularity, and how does it depend on the number \(d\) of dimensions? Such questions are related to that of understanding the asymptotics of \(c_ n\) as \(n\to\infty\). Specifically, does there exist \(\gamma=\gamma(d)\) such that \(c_ n\sim An^{\gamma-1}\mu^ n\) as \(n\to\infty\)? Such a \(\gamma\) is one of a family of ‘critical exponents’ associated with SAW’s. As in the case for other processes, there is a ‘scaling theory’ which makes predictions about the relations between these exponents.

A principal part of the volume is devoted to the theory of the lace expansion in the context of SAW’s. Chapters 5 and 6 contain a careful account of the Hara-Slade theorem on mean-field behaviour of SAW’s when \(d\geq 5\) (details are given for the ‘spread-out’ model, as well as for the nearest-neighbour model for large \(d)\). This conclusion, obtained via the ‘bubble condition’, implies the existence of certain critical exponents, as well as their numerical values. The work of these two chapters is of major current significance. Chapter 9 is a lengthy account of Monte Carlo techniques used to obtain numerical estimates of connective constants, critical exponents, and other numerical values. This account should be very useful to researchers on SAW’s and similar processes. Other chapters contain detailed descriptions of properties of the two-point function, Kesten’s pattern theorem and ratio limit theorems, as well as results for polygons, bridges, and knots. Of course, SAW’s are just the introduction to the general topic of repulsive and/or attractive walks, a hard and broad topic of enormous contemporary interest and importance. Some such relatives of the SAW are mentioned in Chapter 10.

This volume is, together with the recent book of G. F. Lawler [Intersections of random walks (Birkhäuser, 1991)], one of the two landmarks of the modern rigorous theory of walks on lattices.

For simplicity, consider SAW’s on the \(d\)-dimensional hypercubic lattice \(\mathbb{Z}^ d\), where \(d\geq 2\). The two basic questions are

(a) how many SAW’s of length \(n\) exist,

(b) how far apart (on the average) are the endpoints of such a SAW?

In a famous paper, J. M. Hammersley and J. G. Morton [J. Roy. Stat. Soc., Ser. B 16, 23-38 (1954; Zbl 0055.369)] showed that the number \(c_ n\) of such walks satisfies \(c_{m+n}\leq c_ mc_ n\), whence by subadditivity the connective constant \(\mu=\lim_{n\to\infty}c_ n^{1/n}\) exists. The power series \(\chi(z)=\sum_ nc_ nz^ n\) has radius of convergence \(z_ c=\mu^{-1}\), and \(z_ c\) plays the role of the critical point of the model (akin to the critical temperature of a model of statistical mechanics). A principle target is to understand the behaviour of \(\chi\) and other related power series in the vicinity of the singularity \(z_ c\): what is the nature of this singularity, and how does it depend on the number \(d\) of dimensions? Such questions are related to that of understanding the asymptotics of \(c_ n\) as \(n\to\infty\). Specifically, does there exist \(\gamma=\gamma(d)\) such that \(c_ n\sim An^{\gamma-1}\mu^ n\) as \(n\to\infty\)? Such a \(\gamma\) is one of a family of ‘critical exponents’ associated with SAW’s. As in the case for other processes, there is a ‘scaling theory’ which makes predictions about the relations between these exponents.

A principal part of the volume is devoted to the theory of the lace expansion in the context of SAW’s. Chapters 5 and 6 contain a careful account of the Hara-Slade theorem on mean-field behaviour of SAW’s when \(d\geq 5\) (details are given for the ‘spread-out’ model, as well as for the nearest-neighbour model for large \(d)\). This conclusion, obtained via the ‘bubble condition’, implies the existence of certain critical exponents, as well as their numerical values. The work of these two chapters is of major current significance. Chapter 9 is a lengthy account of Monte Carlo techniques used to obtain numerical estimates of connective constants, critical exponents, and other numerical values. This account should be very useful to researchers on SAW’s and similar processes. Other chapters contain detailed descriptions of properties of the two-point function, Kesten’s pattern theorem and ratio limit theorems, as well as results for polygons, bridges, and knots. Of course, SAW’s are just the introduction to the general topic of repulsive and/or attractive walks, a hard and broad topic of enormous contemporary interest and importance. Some such relatives of the SAW are mentioned in Chapter 10.

This volume is, together with the recent book of G. F. Lawler [Intersections of random walks (Birkhäuser, 1991)], one of the two landmarks of the modern rigorous theory of walks on lattices.

Reviewer: G.Grimmett (Cambridge)

##### MSC:

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

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

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |

05C38 | Paths and cycles |