"Self-avoiding walks (SAW)"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 7번째 줄: | 7번째 줄: | ||
==basics== | ==basics== | ||
;def | ;def | ||
| − | A SAW of length $ | + | 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 | ||
| + | * $R_e^2(w)=|w(n)-w(0)|^2$ | ||
| + | * $\langle R_e^2 \rangle=\frac{1}{C_n}\sum_{w\in W_n}R_e^2(w)$ | ||
;conjecture | ;conjecture | ||
| − | + | We have the following conjecture | |
$$ | $$ | ||
| − | + | C_n \sim An^{\gamma-1}\mu^n \\ | |
| + | C_n(x) \sim Bn^{\alpha-2}\mu^n \\ | ||
| + | \langle R_e^2 \rangle \sim Dn^{2\nu} | ||
$$ | $$ | ||
| − | * | + | * universal constant |
| − | ** $ | + | ** $\alpha$ specific heat |
| − | ** $\gamma$ | + | ** $\gamma$ susceptibility |
** $nu$ associated with correlation length | ** $nu$ associated with correlation length | ||
2015년 1월 22일 (목) 06:53 판
introduction
- choose edge in a given lattice
- not allowed to retrace your path
- how many SAWs of length $n$ are there?
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
- $R_e^2(w)=|w(n)-w(0)|^2$
- $\langle R_e^2 \rangle=\frac{1}{C_n}\sum_{w\in W_n}R_e^2(w)$
- conjecture
We have the following conjecture $$ C_n \sim An^{\gamma-1}\mu^n \\ C_n(x) \sim Bn^{\alpha-2}\mu^n \\ \langle R_e^2 \rangle \sim Dn^{2\nu} $$
- universal constant
- $\alpha$ specific heat
- $\gamma$ susceptibility
- $nu$ associated with correlation length
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
- 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.