Hammersley: Crystals


“We deal with atoms and bonds: a bond is a (perhaps directed) path between two atoms. An n-stepped walk is an ordered connected path along n consecutive bonds, each step being in the permitted direction of a bond (if the bond in question is directed) and starting from the atom reached by the previous step (if any). Wn(A) denotes a typical n-stepped walk starting from an atom A. A walk is self-avoiding if it visits no atom more than once. Sn(A) denotes a self-avoiding Wn(A). Two walks are distinct if they do not utilize the same set of bonds, with due regard to order. fA(n, r) denotes the number of distinct Wn(A), each of which can be broken into r or fewer self-avoiding subwalks. In particular we write fA(n) =fA(n, 1) for the number of distinct Sn(A). Two atoms A and B are outlike if fA(n) = fB{n) for all n. An outlike class is a class of pairwise outlike atoms. A crystal is an infinite set of atoms and bonds satisfying the three postulates” (p. 642) #Hammersley #atom #bond #walk #crystal

Hammersley, John M., Percolation processes. II. The Connective Constant, in: Mathematical Proceedings of the Cambridge Philosophical Society 53 (1957), 642–645.

Grimmett: Percolation


“Suppose we immerse a large porous stone in a bucket of water. What is the probability that the centre of the stone is wetted? In formulating a simple stochastic model for such a situation, Broadbent and Hammersley (1957) gave birth to the ‚percolation model‘.” (p. 1) #Grimmett #Broadbent #Hammersley #percolation

Grimmett, Geoffrey R., Percolation. With 76 Illustrations. New York: Springer 1989. 296 S., ISBN 978-1-4757-4210-7.

Broadbent/Hammersley: Self-avoiding Walk


“The fluid will be able to flow from one point to another if and only if there is a connexion without dams between them, and this will be so if and only if there is an undammed self-avoiding walk connecting them (i.e. a walk which visits no intermediate point more than once). It is, therefore, appropriate to study the self-avoiding walks in crystals.” (p. 631) #Broadbent #Hammersley #Self-avoidingWalk

Broadbent, Simon R./Hammersley, John M., Percolation processes. I. Crystals and Mazes, in: Mathematical Proceedings of the Cambridge Philosophical Society 53 (1957), 629–641.

Hammersley: Percolation and Diffusion Processes


“A percolation process is the spread of a fluid through a medium under the influence of a random mechanism associated with the medium. This contrasts with a diffusion process, where the random mechanism is associated with the fluid.” (p. 790) #Hammersley #percolation #diffusion #process

Hammersley, John M., Percolation Processes: Lower Bounds for the Critical Probability, in: The Annals of Mathematical Statistics 28 (1957), 790–795.