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.