"Self-avoiding walks (SAW)"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (Pythagoras0 사용자가 Self-avoiding walks(SAW) 문서를 Self-avoiding walks (SAW) 문서로 옮겼습니다.) |
imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
| − | + | * choose edge in a given lattice | |
| − | + | * not allowed to retrace your path | |
| − | + | * how many SAWs of length $n$ are there? | |
| − | + | ;thm | |
| − | + | 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 | |
| − | + | * the critical exponent $\gamma$ is universal | |
| + | * proof uses discrete holomorphic observables | ||
| 24번째 줄: | 25번째 줄: | ||
* http://en.wikipedia.org/wiki/Self-avoiding_walk | * http://en.wikipedia.org/wiki/Self-avoiding_walk | ||
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==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. | ||
| + | * 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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| − | + | ==encyclopedia== | |
| − | + | * http://en.wikipedia.org/wiki/Connective_constant | |
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:integrable systems]] | [[분류:integrable systems]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
2015년 1월 8일 (목) 15:35 판
introduction
- choose edge in a given lattice
- not allowed to retrace your path
- how many SAWs of length $n$ are there?
- thm
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
- the critical exponent $\gamma$ is universal
- proof uses discrete holomorphic observables
encyclopedia
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.