Self-avoiding walks (SAW)

introduction

• choose edge in a given lattice
• not allowed to retrace your path
• how many SAWs of length $n$ are there?

basics

def

A SAW of length $N$ is a map $W:\{0,1,\cdots, N\} \to \mathbb{Z}^d$ such that $|W(i+1)-W(i)|=1$ and $W(i)\neq W(j)$ for $i\neq j$

• $C_N(x)=C_N$

conjecture

Let $c_n$ be the number of SAWs from a fixed starting point on $\mathbb{Z}^d$. Then $$ c_n \sim An^{\gamma-1}\mu^n $$

• we call
  • $A$ amplitude
  • $\gamma$ : susceptibility
  • $nu$ associated with correlation length

SAW on 2d honeycomb lattice

conjecture

Let $c_n$ be the number of SAWs from a fixed starting point on the honeycomb lattice. Then $$ c_n \sim An^{\gamma-1}\mu^n $$ as $n\to \infty$, where $\mu=\sqrt{2+\sqrt{2}}$ and $\gamma$ is conjectured to be $43/32$

• the fact $\mu=\sqrt{2+\sqrt{2}}$ was conjectured by Nieuhuis in 1982 and proved in 2012 by Smirnov
• the critical exponent $\gamma$ is universal 
• proof uses discrete holomorphic observables

 

related items

• non-intersecting paths

 

computational resource

• https://oeis.org/A001411

 

expositions

• Slade, Gordon. “Self-Avoiding Walks.” The Mathematical Intelligencer 16, no. 1 (December 1, 1994): 29–35. doi:10.1007/BF03026612.

 

articles

• Duminil-Copin, Hugo, and Stanislav Smirnov. “The Connective Constant of the Honeycomb Lattice Equals $\sqrt{2+\sqrt2}$.” arXiv:1007.0575 [math-Ph], July 4, 2010. http://arxiv.org/abs/1007.0575.
• Lawler, Gregory F., Oded Schramm, and Wendelin Werner. “On the Scaling Limit of Planar Self-Avoiding Walk.” arXiv:math/0204277, April 23, 2002. http://arxiv.org/abs/math/0204277.

encyclopedia

• http://en.wikipedia.org/wiki/Connective_constant
• http://en.wikipedia.org/wiki/Self-avoiding_walk