"Self-avoiding walks (SAW)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 3명의 중간 판 33개는 보이지 않습니다) | |||
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| − | < | + | ==introduction== |
| + | * choose edge in a given lattice | ||
| + | * not allowed to retrace your path | ||
| + | * how many SAWs of length <math>n</math> 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 <math>n</math> is a map <math>w:\{0,1,\cdots, n\} \to \mathbb{Z}^d</math> such that <math>|w(i+1)-w(i)|=1</math> and <math>w(i)\neq w(j)</math> for <math>i\neq j</math> | ||
| + | * <math>W_n</math> the set of all SAWs of length <math>n</math> | ||
| + | * <math>C_n(x)=C_n(0,x)</math> number of SAW starting at 0 and ending at x | ||
| + | * <math>C_n=\sum_{x\in \mathbb{Z}^d}C_n(x)</math> number of SAW | ||
| + | * <math>R_e^2(w)=|w(n)-w(0)|^2</math> | ||
| + | * we have | ||
| + | :<math> | ||
| + | \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} | ||
| + | </math> | ||
| + | ;conjecture | ||
| + | We have the following conjecture | ||
| + | :<math> | ||
| + | C_n \sim An^{\gamma-1}\mu^n \label{asymp} | ||
| + | </math> | ||
| + | :<math> | ||
| + | C_n(x) \sim Bn^{\alpha-2}\mu^n | ||
| + | </math> | ||
| + | :<math> | ||
| + | \langle R_e^2 \rangle \sim Dn^{2\nu} | ||
| + | </math> | ||
| + | * critical exponent (universal) | ||
| + | ** <math>\alpha</math> specific heat | ||
| + | ** <math>\gamma</math> susceptibility | ||
| + | ** <math>\nu</math> 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 <math>D</math>-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 <math>d=3</math> | ||
| + | * <math>d\geq 5</math> has been shown that <math>\gamma=1</math> via the lace expansion | ||
| + | * <math>d=4</math> some things proven via exact renormalization group | ||
| + | * <math>d=2</math>, nothing yet, chance of a proof via discrete holomophicity | ||
| + | ==2d lattice== | ||
| + | ===SAW on 2d square lattice=== | ||
| + | * <math>\{c_n\}_{n \geq 0} : 4,12,36,100,\cdots </math> | ||
| + | ===SAW on 2d honeycomb lattice=== | ||
| + | ;conjecture | ||
| + | Let <math>c_n</math> be the number of SAWs from a fixed starting point on the honeycomb lattice. Then | ||
| + | :<math> | ||
| + | c_n \sim An^{\gamma-1}\mu^n | ||
| + | </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 critical exponent <math>\gamma</math> 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 <math>\sqrt{2+\sqrt2}</math>.” 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/Self-avoiding_walk | |
| − | + | * http://en.wikipedia.org/wiki/Connective_constant | |
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| − | + | [[분류:개인노트]] | |
| + | [[분류:integrable systems]] | ||
| + | [[분류:math and physics]] | ||
| + | [[분류:migrate]] | ||
| − | * [ | + | ==메타데이터== |
| − | * [ | + | ===위키데이터=== |
| − | * [ | + | * ID : [https://www.wikidata.org/wiki/Q7448025 Q7448025] |
| − | + | ===Spacy 패턴 목록=== | |
| + | * [{'LOWER': 'self'}, {'OP': '*'}, {'LOWER': 'avoiding'}, {'LEMMA': 'walk'}] | ||
| + | * [{'LEMMA': 'SAW'}] | ||
2021년 2월 17일 (수) 03:15 기준 최신판
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.
encyclopedia
메타데이터
위키데이터
- ID : Q7448025
Spacy 패턴 목록
- [{'LOWER': 'self'}, {'OP': '*'}, {'LOWER': 'avoiding'}, {'LEMMA': 'walk'}]
- [{'LEMMA': 'SAW'}]