Progress in high-dimensional percolation and random graphs
This is a textbook on percolation. The authors consider a lattice -- let us focus on \(\mathbb{Z}^{d}\) here -- where each vertex is adjacent to each of the \(2d\) vertices which differ from it in one coordinate only and then by \(\pm 1\). Percolation is basically the situation where each bond (i.e., edge) is occupied with probability \(p\) and vacant with probability \(1-p\) independent of all other bonds and we study the behaviour of the occupied bonds. For example, for what values of \(p\) is there an infinite cluster of occupied bonds? Letting \(\mathcal{C}(x)\) denote the cluster of a vertex \(x\), we let \(\theta(p)=\mathbb{P}_{p}\left(\vert\mathcal{C}(x)\vert=\infty\right)\). This is clearly independent of \(x\) by transitivity but dependent on \(p\). We let \(p_{c}=\inf \{p: \theta(p)>0\}\) be the critical probability where there is a change from finite clusters to (at least) one infinite cluster. Much of the interest is in values of \(p\) close to \(p_{c}\). For example, a famous question, unsolved in general, is to show that, when \(d\geq 2\), there is no infinite cluster when \(p=p_{c}\). One of the main new contributions of the book under review is to give a self-contained proof that for large enough \(d\) this statement is true (the corresponding result for \(d=2\) is due to \textit{H. Kesten} [Probab. Theory Relat. Fields 73, 369--394 (1986; Zbl 0584.60098)]). It is expected from physical considerations that \(\theta(p)\sim (p-p_{c})^{\beta}\) for some constant \(\beta\) as \(p\downarrow p_{c}\). We might also be interested in \(\chi(p)\), the expected size of the cluster of the origin. Again, physics predicts (but does not prove) the existence of a constant \(\gamma\) such that \(\chi(p)\sim (p-p_{c})^{-\gamma}\). We could further consider \(\chi^{f}(p)\), the size of the cluster of the origin conditional on its being finite, where again it is predicted that \(\chi^{f}(p)\sim (p-p_{c})^{-\delta}\) . (Soma caution in needed about the multiple interpretations of \(\sim\) in this area, see the text for details) . The more detailed description of the first main aim of this book as outlined in the previous paragraph is to prove that in high dimensions we have that \(\beta, \gamma, \delta\) (and another more technical exponent \(\eta\)) exist and take the values \(\beta=\gamma=1\), \(\delta=2\) (and \(\eta=0\)). A secondary main aim is to prove several other, more technical, results on percolation in high dimensions. We do not enumerate these in detail. Here is a somewhat complementary summary of the contents. The case of percolation on a tree is discussed first to help fix ideas, as the lack of cycles makes some aspects technically easier. Chapter 3 proves that \(p_{c}\) is equal to another key probability \(p_{T}=\sup \{p: \chi(p)<\infty\}\). (That \(p_{T}\leq p_{c}\) is obvious, the work is in the other direction). Part II proves the ``mean-field behaviour'' in high dimensions, including the results mentioned above about exponents: key concepts here include the so-called triangle condition, and the infrared bound which leads to a proof that the triangle condition holds in high dimensions, the main tool in the proof of the infrared bound being the so-called lace expansion which ultimately relies on inclusion-exclusion. So far the aim has been for the text to be self-contained, but Part III on some other recent results in high-dimensional percolation and Part IV on some related models are given with somewhat less detailed proofs. Many open problems are also presented.
scientific article; zbMATH DE number 6755664