"Self-avoiding walks (SAW)"의 두 판 사이의 차이
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as <math>n\to \infty</math>, where <math>\mu=\sqrt{2+\sqrt{2}}</math> and <math>\gamma</math> is conjectured to be <math>43/32</math> | as <math>n\to \infty</math>, where <math>\mu=\sqrt{2+\sqrt{2}}</math> and <math>\gamma</math> is conjectured to be <math>43/32</math> | ||
* the fact <math>\mu=\sqrt{2+\sqrt{2}}</math> was conjectured by Nieuhuis in 1982 and proved in 2012 by Smirnov | * the fact <math>\mu=\sqrt{2+\sqrt{2}}</math> was conjectured by Nieuhuis in 1982 and proved in 2012 by Smirnov | ||
| − | * the critical exponent <math>\gamma</math> is | + | * the critical exponent <math>\gamma</math> is universal |
* proof uses discrete holomorphic observables | * proof uses discrete holomorphic observables | ||
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==related items== | ==related items== | ||
| 79번째 줄: | 79번째 줄: | ||
* [[non-intersecting paths]] | * [[non-intersecting paths]] | ||
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==computational resource== | ==computational resource== | ||
* https://oeis.org/A001411 | * https://oeis.org/A001411 | ||
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| 89번째 줄: | 89번째 줄: | ||
* Slade, Gordon. “Self-Avoiding Walks.” The Mathematical Intelligencer 16, no. 1 (December 1, 1994): 29–35. doi:10.1007/BF03026612. | * Slade, Gordon. “Self-Avoiding Walks.” The Mathematical Intelligencer 16, no. 1 (December 1, 1994): 29–35. doi:10.1007/BF03026612. | ||
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==articles== | ==articles== | ||
2020년 12월 28일 (월) 04:05 판
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
- Coulomb 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.