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

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2번째 줄: 2번째 줄:
 
* choose edge in a given lattice
 
* choose edge in a given lattice
 
* not allowed to retrace your path
 
* not allowed to retrace your path
−
* how many SAWs of length $n$ are there?
+
* 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
 
* simple to define, in some ways really easy to study but we are not close to a closed form formula
  
 
==basics==
 
==basics==
 
;def
 
;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 $i\neq j$
+
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>
−
* $W_n$ the set of all SAWs of length $n$
+
* <math>W_n</math> the set of all SAWs of length <math>n</math>
−
* $C_n(x)=C_n(0,x)$ number of SAW starting at 0 and ending at x
+
* <math>C_n(x)=C_n(0,x)</math> number of SAW starting at 0 and ending at x
−
* $C_n=\sum_{x\in \mathbb{Z}^d}C_n(x)$ number of SAW
+
* <math>C_n=\sum_{x\in \mathbb{Z}^d}C_n(x)</math> number of SAW
−
* $R_e^2(w)=|w(n)-w(0)|^2$
+
* <math>R_e^2(w)=|w(n)-w(0)|^2</math>
 
* we have
 
* we have
−
$$
+
:<math>
 
\begin{align}
 
\begin{align}
 
\langle R_e^2 \rangle&=\frac{1}{C_n}\sum_{w\in W_n}R_e^2(w) \\
 
\langle R_e^2 \rangle&=\frac{1}{C_n}\sum_{w\in W_n}R_e^2(w) \\
20번째 줄: 20번째 줄:
 
&=\frac{1}{C_n}\sum_{x\in \mathbb{Z}^d}|x|^2C_n(x)
 
&=\frac{1}{C_n}\sum_{x\in \mathbb{Z}^d}|x|^2C_n(x)
 
\end{align}
 
\end{align}
−
$$
+
</math>
 
;conjecture  
 
;conjecture  
 
We have the following conjecture
 
We have the following conjecture
−
$$
+
:<math>
 
C_n \sim An^{\gamma-1}\mu^n \label{asymp}
 
C_n \sim An^{\gamma-1}\mu^n \label{asymp}
−
$$
+
</math>
−
$$
+
:<math>
 
C_n(x) \sim Bn^{\alpha-2}\mu^n
 
C_n(x) \sim Bn^{\alpha-2}\mu^n
−
$$
+
</math>
−
$$
+
:<math>
 
\langle R_e^2 \rangle \sim Dn^{2\nu}
 
\langle R_e^2 \rangle \sim Dn^{2\nu}
−
$$
+
</math>
 
* critical exponent (universal)
 
* critical exponent (universal)
−
** $\alpha$ specific heat
+
** <math>\alpha</math> specific heat
−
** $\gamma$ susceptibility
+
** <math>\gamma</math> susceptibility
−
** $\nu$ associated with correlation length
+
** <math>\nu</math> associated with correlation length
 
==models in the universality class==
 
==models in the universality class==
 
* Domb-Joyce : weakly avoiding walk (penalty for intersection)
 
* Domb-Joyce : weakly avoiding walk (penalty for intersection)
42번째 줄: 42번째 줄:
  
 
==overview of known results==
 
==overview of known results==
−
* any solution will not be $D$-finite
+
* any solution will not be <math>D</math>-finite
 
===2d===
 
===2d===
−
* Columb gas (early 1980's)
+
* Coulomb gas (early 1980's)
 
* conformal field theory (1980's)
 
* conformal field theory (1980's)
 
* SLE (since 1998)
 
* SLE (since 1998)
56번째 줄: 56번째 줄:
  
 
===asymptotics \ref{asymp}===
 
===asymptotics \ref{asymp}===
−
* very little hope of showing this in $d=3$
+
* very little hope of showing this in <math>d=3</math>
−
* $d\geq 5$ has been shown that $\gamma=1$ via the lace expansion
+
* <math>d\geq 5</math> has been shown that <math>\gamma=1</math> via the lace expansion
−
* $d=4$ some things proven via exact renormalization group
+
* <math>d=4</math> some things proven via exact renormalization group
−
* $d=2$, nothing yet, chance of a proof via discrete holomophicity
+
* <math>d=2</math>, nothing yet, chance of a proof via discrete holomophicity
−
 
+
==2d lattice==
−
==SAW on 2d honeycomb lattice==
+
===SAW on 2d square lattice===
 +
* <math>\{c_n\}_{n \geq 0} : 4,12,36,100,\cdots </math>
 +
===SAW on 2d honeycomb lattice===
 
;conjecture  
 
;conjecture  
−
Let $c_n$ be the number of SAWs from a fixed starting point on the honeycomb lattice. Then
+
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
 
c_n \sim An^{\gamma-1}\mu^n
−
$$
+
</math>
−
as $n\to \infty$, where $\mu=\sqrt{2+\sqrt{2}}$ and $\gamma$ is conjectured to be $43/32$
+
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 $\mu=\sqrt{2+\sqrt{2}}$ 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 $\gamma$ is universal 
+
* the critical exponent <math>\gamma</math> is universal
 
* proof uses discrete holomorphic observables
 
* proof uses discrete holomorphic observables
−
 
+
  
 
==related items==
 
==related items==
77번째 줄: 79번째 줄:
 
* [[non-intersecting paths]]
 
* [[non-intersecting paths]]
  
−
 
+
 
==computational resource==
 
==computational resource==
 
* https://oeis.org/A001411
 
* https://oeis.org/A001411
−
 
+
  
  
87번째 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.
  
−
 
+
  
 
==articles==
 
==articles==
−
* 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.
+
* 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.
 
* 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==
 
==encyclopedia==
 +
* http://en.wikipedia.org/wiki/Self-avoiding_walk
 
* http://en.wikipedia.org/wiki/Connective_constant
 
* http://en.wikipedia.org/wiki/Connective_constant
−
* http://en.wikipedia.org/wiki/Self-avoiding_walk
 
  
  
102번째 줄: 104번째 줄:
 
[[분류:integrable systems]]
 
[[분류:integrable systems]]
 
[[분류:math and physics]]
 
[[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) 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"