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
computational resource
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.