2.8 Historical Notes and Further Reading

Theorem 2.1.1 is a classic result from branching processes. These date back to the work of Watson and Galton (1874) on the survival of surnames in the British peerage. A classic book on the subject is the one by Harris (1963). Theorem 2.2.5 dates back to Broadbent and Hammersley s (1957) original paper on discrete percolation. A general reference for discrete percolation models is the book by Grimmett (1999). Theorem 2.2.7 appears in Grimmett and Stacey (1998), where the strict inequality for a broad range of graphs is also shown. A stronger version of Theorem 2.3.1 can be found in Liggett, Schonmann, and Stacey (1997), who show that percolation occurs as long as the occupation probability of any site (edge), conditioned to the states of all sites (edges) outside a finite neighbourhood of it, can be made sufficiently high. Theorem 2.4.1 is by H ggstr m and Meester (1996). Theorem 2.5.1 is by Penrose (1991). Theorems 2.5.2, 2.5.4, 2.5.5 are by Franceschetti, Booth et al. (2005). Theorem 2.5.4 was also discovered independently by Balister, Bollob s, and Walters (2004). Similar results, for a different spreading transformation, appear in Penrose (1993), and Meester, Penrose, and Sarkar (1997). Theorem 2.5.6 is by Penrose (1991), while compression results for the boolean model appear in Alexander (1991). Theorem 2.6.1 dates back to Gilbert (1961). A general reference for continuum models is the book by Meester and Roy (1996). The treatment of the signal to noise plus interference model follows...