In computational physics a self-avoiding walk is a chain-like path in **R**^{2} or **R**^{3} with a certain number of nodes, typically a fixed step length and has the imperative property that it doesn't cross itself or another walk. A system of self-avoiding walks satisfies the so-called excluded volume condition. In higher dimensions, the self-avoiding walk is believed to behave much like the ordinary random walk. SAWs and SAPs play a central role in the modelling of the topological and knot-theoretic behaviour of thread- and loop-like molecules such as proteins. SAW is a fractal.^{}^{} For example, in *d* = 2 the fractal dimension is 4/3, for *d* = 3 it is close to 5/3 while for *d* ≥ 4 the fractal dimension is 2. The dimension is called the upper critical dimension above which excluded volume is negligible. A SAW that does not satisfy the excluded volume condition was recently studied to model explicit surface geometry resulting from expansion of a SAW.^{}

The properties of SAWs cannot be calculated analytically, so numerical simulations are employed. The pivot algorithm is a common method for Markov chain Monte Carlo simulations for the uniform measure on n-step self-avoiding walks. The pivot algorithm works by taking a self-avoiding walk and randomly choosing a point on this walk, and then applying a symmetry operation (rotations and reflections) on the walk after the nth step to create a new walk. Calculating the number of self-avoiding walks in any given lattice is a common computational problem. There is currently no known formula for determining the number of self-avoiding walks, although there are rigorous methods for approximating them.^{}^{} Finding the number of such paths is conjectured to be an NP-hard problem^{}. For self-avoiding walks from one end of a diagonal to the other, with only moves in the positive direction, there are exactly

paths for an *m* × *n* rectangular lattice.

One of the phenomena associated with self-avoiding walks and 2-dimensional statistical physics models in general is the notion of universality, that is, independence of macroscopic observables from microscopic details, such as the choice of the lattice. One important quantity that appears in conjectures for universal laws is the connective constant, defined as follows. Let c_{n} denote the number of n-step self-avoiding walks. Since every (*n* + *m*)-step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that *c*_{n+m} ≤ *c _{n}c_{m}*. Therefore, the sequence {log

μ is called the **connective constant**, since c_{n} depends on the particular lattice chosen for the walk so does μ. The exact value of μ is only known for the hexagonal lattice, where it is equal to:^{}

This page was last edited on 20 December 2017, at 23:27.

Reference: https://en.wikipedia.org/wiki/Self-avoiding_walk under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Self-avoiding_walk under CC BY-SA license.

- Mathematics
- Sequence
- Lattice
- Lattice Path
- Graph Theoretical
- Path
- Paul Flory
- Solvents
- Polymers
- Computational Physics
- Excluded Volume
- Random Walk
- Topological
- Knot-theoretic
- Proteins
- Fractal
- Fractal Dimension
- Critical Dimension
- Simulations
- Pivot Algorithm
- Markov Chain Monte Carlo
- Conjectured
- NP-hard
- Connective Constant
- Subadditive
- Fekete's Lemma

- New Technology Train
- Self-avoiding Walk
- Kolderwolde
- Olympic Gods
- Encelia Virginensis
- Logie Awards Of 1990
- Little A
- South Central Coast
- Queen Paola Of Belgium
- Stethoscope
- Pauli Matrix
- Belly Dancer
- Project Tiger
- Iconic Memory
- Open Cobalt
- District Electoral Division
- Glymphatic System
- Declaration Of Philadelphia
- Metternich System
- Nike Zeus
- I Would Do Anything For Love (But I Won%27t Do That)
- Frank K. Richardson
- A. B. Facey
- Advowsons
- Ebenezer J. Hill
- Emerson C. Angell
- Joel R. P. Pringle
- James W. Symington
- John S. Fullmer
- Addison C. Gibbs