"Bootstrap percolation"의 두 판 사이의 차이

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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">introduction==
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==introduction==
  
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*  one of important question in 2d percolation is the calculation of power-law exponent for boostrap percolation<br>
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*  one of important question in 2d percolation is the calculation of power-law exponent for boostrap percolation
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*  this is related to the theory of partitions without k-gaps<br>  <br>
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*  this is related to the theory of partitions without k-gaps
  
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">bootstrap percolation==
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==bootstrap percolation==
  
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*  growth rule<br>
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*  growth rule
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* http://www.math.ubc.ca/~holroyd/boot/<br>
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* http://www.math.ubc.ca/~holroyd/boot/
  
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* http://mathworld.wolfram.com/BootstrapPercolation.html<br>
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* http://mathworld.wolfram.com/BootstrapPercolation.html
  
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<h5 style="line-height: 2em; margin: 0px;">partitions without k-gaps==
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==partitions without k-gaps==
  
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*  partitions without k-gaps (or k-sequences)<br>
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*  partitions without k-gaps (or k-sequences)
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*  p_k(n) is the number of partitions of n that do not contain any sequence of consecutive integers of length k. p_2 (7) = 8.<br>
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*  p_k(n) is the number of partitions of n that do not contain any sequence of consecutive integers of length k. p_2 (7) = 8.
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*  examples: partition of 7<br> {{7},{6,1},{5,2},{5,1,1},{4,3},{4,2,1},{4,1,1,1},{3,3,1},{3,2,2},{3,2,1,1},{3,1,1,1,1},{2,2,2,1},{2,2,1,1,1},{2,1,1,1,1,1},{1,1,1,1,1,1,1}}<br> 7, 6 + 1, 5 + 2, 5 + 1 + 1, 4 + 1 + 1 + 1, 3 + 3 + 1, 3 + 1 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1 + 1 + 1.<br> so there are 8 partitions without 2-gaps<br>
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*  examples: partition of 7 {{7},{6,1},{5,2},{5,1,1},{4,3},{4,2,1},{4,1,1,1},{3,3,1},{3,2,2},{3,2,1,1},{3,1,1,1,1},{2,2,2,1},{2,2,1,1,1},{2,1,1,1,1,1},{1,1,1,1,1,1,1}} 7, 6 + 1, 5 + 2, 5 + 1 + 1, 4 + 1 + 1 + 1, 3 + 3 + 1, 3 + 1 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1 + 1 + 1. so there are 8 partitions without 2-gaps
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*  Anderew's result<br>
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*  Anderew's result
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**  generating function for partitions without k-gaps<br><math>G_2(q)=1+\sum_{n=1}^{\infty}\frac{q^n\prod_{j=1}^{n-1}(1-q^j+q^{2j})}{(q;q)_n}</math>[http://www.research.att.com/%7Enjas/sequences/A116931 A116931]<br>
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**  generating function for partitions without k-gaps<math>G_2(q)=1+\sum_{n=1}^{\infty}\frac{q^n\prod_{j=1}^{n-1}(1-q^j+q^{2j})}{(q;q)_n}</math>[http://www.research.att.com/%7Enjas/sequences/A116931 A116931]
  
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#  (*define a gap as 'b' *)<br> b := 2<br> G[b_, x_] :=<br>  Sum[x^k*Product[1 + x^(b*j)/(1 - x^j), {j, 1, k - 1}]/(1 - x^k), {k,<br>    1, 30}]<br> Series[G[b, x], {x, 0, 20}]<br> Table[SeriesCoefficient[%, n], {n, 0, 20}]<br>
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#  (*define a gap as 'b' *) b := 2 G[b_, x_] := Sum[x^k*Product[1 + x^(b*j)/(1 - x^j), {j, 1, k - 1}]/(1 - x^k), {k,   1, 30}] Series[G[b, x], {x, 0, 20}] Table[SeriesCoefficient[%, n], {n, 0, 20}]
  
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* [[3 q-series|q-series]]<br>
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* [[3 q-series|q-series]]
  
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<h5 style="line-height: 2em; margin: 0px;">q-series from percolation==
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==q-series from percolation==
  
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*  definition<br><math>P_k(q)=(q;q)_{\infty}G_k(q)</math><br>
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*  definition<math>P_k(q)=(q;q)_{\infty}G_k(q)</math>
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*  Andrews and Zagier expression of <math>P_k(q)</math><br>
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*  Andrews and Zagier expression of <math>P_k(q)</math>
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*  result of '''[HLR04]'''<br> if <math>q=e^{-t}</math> and  <math>t\sim 0</math><br><math>P_k(q) \sim \frac{-\lambda_k}{1-q}</math> as <math>q \to 1</math><br>
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*  result of '''[HLR04]''' if <math>q=e^{-t}</math> and  <math>t\sim 0</math><math>P_k(q) \sim \frac{-\lambda_k}{1-q}</math> as <math>q \to 1</math>
  
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<h5 style="line-height: 2em; margin: 0px;">Andrews' conjecture on asymptotics==
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==Andrews' conjecture on asymptotics==
  
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*  asymptotics of P_2(q) is known <br><math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})</math><br>
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*  asymptotics of P_2(q) is known <math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<math>P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})</math>
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*  conjecture<br><math>P_k(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\lambda_k}{t})</math><br> where <math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math><br>
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*  conjecture<math>P_k(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\lambda_k}{t})</math> where <math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math>
  
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">tricky integrals==
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==tricky integrals==
  
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*  Henrik Eriksson: [http://www.math.ubc.ca/~holroyd/integral.pdf A Tricky Integral]<br><math>f_1(x)=1-x</math><br><math>f_2(x)=\frac{1-x+\sqrt{(1-x)(1+3x)}}{2}</math><br>
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*  Henrik Eriksson: [http://www.math.ubc.ca/~holroyd/integral.pdf A Tricky Integral]<math>f_1(x)=1-x</math><math>f_2(x)=\frac{1-x+\sqrt{(1-x)(1+3x)}}{2}</math>
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* <math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math><br>
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* <math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math>
 
* <math>\lambda_2=\frac{\pi^2}{18}</math>
 
* <math>\lambda_2=\frac{\pi^2}{18}</math>
  
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<h5 style="line-height: 2em; margin: 0px;">relevance to dedekind eta function==
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==relevance to dedekind eta function==
  
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*  Dedekind eta function ([http://pythagoras0.springnote.com/pages/3325777 데데킨트 에타함수])<br><math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})</math><br> more generally, <math>q=\exp(\frac{2\pi ih}{k})e^{-t}</math>  and  <math>t\to 0</math> implies<br><math>\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim  \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}</math><br>
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*  Dedekind eta function ([http://pythagoras0.springnote.com/pages/3325777 데데킨트 에타함수])<math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<math>\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})</math> more generally, <math>q=\exp(\frac{2\pi ih}{k})e^{-t}</math> and  <math>t\to 0</math> implies<math>\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim  \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}</math>
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">history==
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==related items==
  
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [[asymptotic analysis of basic hypergeometric series]]
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* [[examples of mock theta functions|Ramanujan's mock theta functions]]
  
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==articles==
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">related items==
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* [http://arxiv.org/abs/1002.3881 A sharper threshold for bootstrap percolation in two dimensions]
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* [[asymptotic analysis of basic hypergeometric series]]<br>
 
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* [[examples of mock theta functions|Ramanujan's mock theta functions]]<br>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia==
 
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">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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[[4909919|4909919]]
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles==
 
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* [http://arxiv.org/abs/1002.3881 A sharper threshold for bootstrap percolation in two dimensions]<br>
 
 
** [http://arxiv.org/abs/1002.3881 ]Janko Gravner, Alexander E. Holroyd, Robert Morris, 2010
 
** [http://arxiv.org/abs/1002.3881 ]Janko Gravner, Alexander E. Holroyd, Robert Morris, 2010
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* [http://arxiv.org/abs/1001.1977 Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation]<br>
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* [http://arxiv.org/abs/1001.1977 Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation]
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** [http://arxiv.org/find/math/1/au:+Bringmann_K/0/1/0/all/0/1 Kathrin Bringmann], [http://arxiv.org/find/math/1/au:+Mahlburg_K/0/1/0/all/0/1 Karl Mahlburg], 2010<br>
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** [http://arxiv.org/find/math/1/au:+Bringmann_K/0/1/0/all/0/1 Kathrin Bringmann], [http://arxiv.org/find/math/1/au:+Mahlburg_K/0/1/0/all/0/1 Karl Mahlburg], 2010
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* [http://dx.doi.org/10.1016/j.jcta.2006.06.010 Integrals, partitions and MacMahon's Theorem]<br>
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* [http://dx.doi.org/10.1016/j.jcta.2006.06.010 Integrals, partitions and MacMahon's Theorem]
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**  George Andrewsa,  Dan Romik, 2007<br>
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**  George Andrewsa, Dan Romik, 2007
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*  Slow convergence<br>
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*  Slow convergence
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* [http://www.pnas.org/content/102/13/4666.full Partitions with short sequences and mock theta functions]<br>
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* [http://www.pnas.org/content/102/13/4666.full Partitions with short sequences and mock theta functions]
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**  George E. Andrews, 2005<br>
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**  George E. Andrews, 2005
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* '''[HLR04]'''[http://research.microsoft.com/en-us/um/people/holroyd/papers/int.pdf Integrals, Partitions, and Cellular Automata]<br>
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* '''[HLR04]'''[http://research.microsoft.com/en-us/um/people/holroyd/papers/int.pdf Integrals, Partitions, and Cellular Automata]
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**  A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368<br>
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**  A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368
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* [http://www.springerlink.com/content/g420hc5h6qu6e65x/ sharp metastability threshold for two-dimensional bootstrap percolation]<br>
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* [http://www.springerlink.com/content/g420hc5h6qu6e65x/ sharp metastability threshold for two-dimensional bootstrap percolation]
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**  Alexander E. Holroyd, 2003<br>
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**  Alexander E. Holroyd, 2003
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* [[2010년 books and articles|논문정리]]
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[[분류:integrable systems]]
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* http://www.ams.org/mathscinet
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[[분류:math and physics]]
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* http://www.zentralblatt-math.org/zmath/en/
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[[분류:migrate]]
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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][http://www.ams.org/mathscinet ]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* http://mathoverflow.net/search?q=
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs==
 
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*  구글 블로그 검색<br>
 
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* [http://www.math.ubc.ca/%7Eholroyd/ http://www.math.ubc.ca/~holroyd/]
 
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* http://arxiv.org/
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">TeX ==
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q25305507 Q25305507]
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===Spacy 패턴 목록===
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* [{'LOWER': 'bootstrap'}, {'LEMMA': 'percolation'}]

2021년 2월 17일 (수) 02:59 기준 최신판

introduction

  • one of important question in 2d percolation is the calculation of power-law exponent for boostrap percolation
  • this is related to the theory of partitions without k-gaps


bootstrap percolation



partitions without k-gaps

  • partitions without k-gaps (or k-sequences)
  • p_k(n) is the number of partitions of n that do not contain any sequence of consecutive integers of length k. p_2 (7) = 8.
  • examples: partition of 7 {{7},{6,1},{5,2},{5,1,1},{4,3},{4,2,1},{4,1,1,1},{3,3,1},{3,2,2},{3,2,1,1},{3,1,1,1,1},{2,2,2,1},{2,2,1,1,1},{2,1,1,1,1,1},{1,1,1,1,1,1,1}} 7, 6 + 1, 5 + 2, 5 + 1 + 1, 4 + 1 + 1 + 1, 3 + 3 + 1, 3 + 1 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1 + 1 + 1. so there are 8 partitions without 2-gaps
  • Anderew's result
    • generating function for partitions without k-gaps\(G_2(q)=1+\sum_{n=1}^{\infty}\frac{q^n\prod_{j=1}^{n-1}(1-q^j+q^{2j})}{(q;q)_n}\)A116931
  1. (*define a gap as 'b' *) b := 2 G[b_, x_] := Sum[x^k*Product[1 + x^(b*j)/(1 - x^j), {j, 1, k - 1}]/(1 - x^k), {k, 1, 30}] Series[G[b, x], {x, 0, 20}] Table[SeriesCoefficient[%, n], {n, 0, 20}]



q-series from percolation

  • definition\(P_k(q)=(q;q)_{\infty}G_k(q)\)
  • Andrews and Zagier expression of \(P_k(q)\)
  • result of [HLR04] if \(q=e^{-t}\) and \(t\sim 0\)\(P_k(q) \sim \frac{-\lambda_k}{1-q}\) as \(q \to 1\)



Andrews' conjecture on asymptotics

  • asymptotics of P_2(q) is known \(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,\(P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})\)
  • conjecture\(P_k(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\lambda_k}{t})\) where \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)



tricky integrals

  • Henrik Eriksson: A Tricky Integral\(f_1(x)=1-x\)\(f_2(x)=\frac{1-x+\sqrt{(1-x)(1+3x)}}{2}\)
  • \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)
  • \(\lambda_2=\frac{\pi^2}{18}\)



relevance to dedekind eta function

  • Dedekind eta function (데데킨트 에타함수)\(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,\(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})\) more generally, \(q=\exp(\frac{2\pi ih}{k})e^{-t}\) and \(t\to 0\) implies\(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)



related items


articles

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'bootstrap'}, {'LEMMA': 'percolation'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Bootstrap_percolation&oldid=51868"