"Self-avoiding walks (SAW)"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 3번째 줄: | 3번째 줄: | ||
* 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 $n$ are there? | ||
| − | ; | + | ;conjecture |
Let $c_n$ be the number of SAWs from a fixed starting point on the honeycomb lattice. Then | Let $c_n$ be the number of SAWs from a fixed starting point on the honeycomb lattice. Then | ||
$$ | $$ | ||
| 9번째 줄: | 9번째 줄: | ||
$$ | $$ | ||
as $n\to \infty$, where $\mu=\sqrt{2+\sqrt{2}}$ and $\gamma$ is conjectured to be $43/32$ | 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 fact $\mu=\sqrt{2+\sqrt{2}}$ was conjectured by Nieuhuis in 1982 and proved in 2012 by Smirnov |
* the critical exponent $\gamma$ is universal | * the critical exponent $\gamma$ is universal | ||
* proof uses discrete holomorphic observables | * proof uses discrete holomorphic observables | ||
2015년 1월 22일 (목) 06:28 판
introduction
- choose edge in a given lattice
- not allowed to retrace your path
- how many SAWs of length $n$ are there?
- 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
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.