of the lattice. One important quantity that appears in conjectures for universal laws is the connective constant, defined as follows. Let denote the number of -step self-avoiding walks. Since every -step self avoiding walk can be decomposed into an -step self-avoiding walk and an -step self-avoiding walk, it follows that . Therefore, the sequence is subadditive and we can apply Fekete's lemma to show that the following limit exists: is called the connective constant, since depends on the particular lattice chosen for the walk so does . The exact value of is only known for the hexagonal lattice, where it is equal to: For other lattices, has only been approximated numerically, and is believed not to even be an algebraic number. It is conjectured that as , where depends on the lattice, but the power law correction does not; in other words, this law is believed to be universal. On networks Self-avoiding walks have also been studied in the context of network theory. In this context, it is customary to treat the SAW as a dynamical process, such that in every time-step a walker randomly hops between neighboring nodes of the network. The walk ends when the walker reaches a dead-end state, such that it can no longer progress to newly un-visited nodes. It was recently found that on Erdős–Rényi networks, the distribution of path lengths of such dynamically grown SAWs can be calculated analytically, and follows the Gompertz distribution. Limits Consider the uniform measure on -step self-avoiding walks in the full plane. It is currently unknown whether the limit of the uniform measure as induces a measure on infinite full-plane walks. However, Harry Kesten has shown that such a measure exists for self-avoiding walks in the half-plane. One important question involving self-avoiding walks is the existence and conformal invariance of the scaling limit, that is, the limit as the length of the walk goes to infinity and the mesh of the lattice goes to zero. The scaling limit of the self-avoiding walk is conjectured to be described by Schramm–Loewner evolution with parameter See also Critical phenomena Hamiltonian path Knight's tour Random walk Snake (video game genre) Universality (dynamical systems) References Further reading External links —the number of self-avoiding paths joining opposite corners of an N × N grid, for N from 0 to 12. Also includes an extended list up to N = 21. Java applet of a 2D