"Self-avoiding walks (SAW)"의 두 판 사이의 차이

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==introduction==
 
==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)
  
−
==history==
+
===3d===
 +
* no exact prediction
 +
* numerical method
 +
* renormalization group
 +
* series method
 +
* monte carlo simultation
  
−
* http://www.google.com/search?hl=en&tbs=tl:1&q=
+
===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==
 
==related items==
17번째 줄: 79번째 줄:
 
* [[non-intersecting paths]]
 
* [[non-intersecting paths]]
  
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==computational resource==
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* https://oeis.org/A001411
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==encyclopedia==
 
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* http://en.wikipedia.org/wiki/Self-avoiding_walk
 
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* http://www.scholarpedia.org/
 
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* http://www.proofwiki.org/wiki/
 
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==books==
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
  
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==expositions==
 
==expositions==
 +
* 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==
 +
* 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.
  
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* [http://arxiv.org/abs/math/0204277 On the scaling limit of planar self-avoiding walk]<br>
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==encyclopedia==
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* http://www.ams.org/mathscinet
+
* http://en.wikipedia.org/wiki/Self-avoiding_walk
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* http://www.zentralblatt-math.org/zmath/en/
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* http://en.wikipedia.org/wiki/Connective_constant
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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==question and answers(Math Overflow)==
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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==blogs==
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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==experts on the field==
 
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* http://arxiv.org/
 
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==links==
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* http://functions.wolfram.com/
 
 
[[분류:개인노트]]
 
[[분류:개인노트]]
 
[[분류:integrable systems]]
 
[[분류:integrable systems]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
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[[분류:migrate]]
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==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) 02:15 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


related items


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

Spacy 패턴 목록

  • [{'LOWER': 'self'}, {'OP': '*'}, {'LOWER': 'avoiding'}, {'LEMMA': 'walk'}]
  • [{'LEMMA': 'SAW'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Self-avoiding_walks_(SAW)&oldid=51945"