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?
• simple to define, in some ways really easy to study but we are not close to a closed form formula

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 $i\neq j$

• $W_n$ the set of all SAWs of length $n$
• $C_n(x)=C_n(0,x)$ number of SAW starting at 0 and ending at x
• $C_n=\sum_{x\in \mathbb{Z}^d}C_n(x)$ number of SAW
• $R_e^2(w)=|w(n)-w(0)|^2$
• we have

$$ \begin{align} \langle R_e^2 \rangle&=\frac{1}{C_n}\sum_{w\in W_n}R_e^2(w) \\ &=\frac{1}{C_n}\sum_{w\in W_n}|w(n)|^2 \\ &=\frac{1}{C_n}\sum_{x\in \mathbb{Z}^d}\sum_{w:w(n)=x}|x|^2 \\ &=\frac{1}{C_n}\sum_{x\in \mathbb{Z}^d}|x|^2C_n(x) \end{align} $$

conjecture

We have the following conjecture $$ C_n \sim An^{\gamma-1}\mu^n \label{asymp} $$ $$ C_n(x) \sim Bn^{\alpha-2}\mu^n $$ $$ \langle R_e^2 \rangle \sim Dn^{2\nu} $$

• critical exponent (universal)
  • $\alpha$ specific heat
  • $\gamma$ susceptibility
  • $\nu$ associated with correlation length

models in the universality class

• Domb-Joyce : weakly avoiding walk (penalty for intersection)
• bead model in the continuum
• polymers

overview of known results

• any solution will not be $D$-finite

2d

• Columb gas (early 1980's)
• conformal field theory (1980's)
• SLE (since 1998)

3d

• no exact prediction
• numerical method
• renormalization group
• series method
• monte carlo simultation

asymptotics \ref{asymp}

• very little hope of showing this in $d=3$
• $d\geq 5$ has been shown that $\gamma=1$ via the lace expansion
• $d=4$ some things proven via exact renormalization group
• $d=2$, nothing yet, chance of a proof via discrete holomophicity

2d lattice

SAW on 2d square lattice

• $\{c_n\}_{n \geq 0} : 4,12,36,100,\cdots $

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

• Grimmett, Geoffrey R., and Zhongyang Li. “Counting Self-Avoiding Walks.” arXiv:1304.7216 [math-Ph], April 26, 2013. http://arxiv.org/abs/1304.7216.
• 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