"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</h5>
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==introduction==
  
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*  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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*  growth rule<br>
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*  this is related to the theory of partitions without k-gaps 
  
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* http://mathworld.wolfram.com/BootstrapPercolation.html<br>
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==bootstrap percolation==
  
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*  growth rule
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* http://www.math.ubc.ca/~holroyd/boot/
  
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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</h5>
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* partitions without k-gaps<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.<br>
 
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*  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> {{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><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><br>[http://www.research.att.com/%7Enjas/sequences/A116931 A116931]<br>
 
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*#  (*define a gap *)<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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* [[3 q-series|q-series]]<br>
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==partitions without k-gaps==
  
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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.
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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
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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' *) 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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<h5 style="line-height: 2em; margin: 0px;">tricky integrals</h5>
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* [[3 q-series|q-series]]
  
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* Henrik Eriksson: [http://www.math.ubc.ca/%7Eholroyd/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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* <math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math><br>
 
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* <math>\lambda_2=\frac{\pi^2}{18}</math>
 
  
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==q-series from percolation==
  
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<h5 style="line-height: 2em; margin: 0px;">q-series</h5>
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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>
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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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* definition<br><math>P_k(q)</math><br>
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*  asymptotics of P_2(q)<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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==Andrews' conjecture on asymptotics==
  
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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<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: 0px;">Andrews' conjecture</h5>
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<math>P_k(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\lambda_k}{t})</math>
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==tricky integrals==
  
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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>
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<h5 style="line-height: 2em; margin: 0px;">relevance to dedekind eta function</h5>
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* <math>\lambda_2=\frac{\pi^2}{18}</math>
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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})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})</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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[http://www-thphys.physics.ox.ac.uk/people/JohnCardy/chuo.ps Lectures on Conformal Invariance and Percolation] Lectures delivered at Chuo University, Tokyo, March 2001. [http://www-thphys.physics.ox.ac.uk/people/JohnCardy/chuo.pdf pdf version]
 
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Conformal Field Theory and Percolation <br><code style="font: 110% Monaco,Courier,monospace;">Authors:</code> Annekathrin Müller-Lohmann <br><code style="font: 110% Monaco,Courier,monospace;">Sources:</code>[http://www.itp.uni-hannover.de/%7Eflohr/papers.html# ]AbstractIn this thesis, important features of two dimensional bond percolation on an infinite square lattice at its critical point within a conformal field theory (CFT) approach are presented. This includes a level three null vector interpretation for Watts' differential equation [[http://arxiv.org/abs/condmat/9603167 78]] describing the horizontal vertical crossing probability within this setup. A unique solution among the minimal models, c<sub style="">(6,1)</sub>= -24, seems to be a good candidate, satisfying the level two differential equation for the horizontal crossing probability derived by Cardy [[http://arxiv.org/abs/hepth/9111026 7]] as well.<br>     Commonly assumed to be a truly scale invariant problem, percolation nevertheless is usually investigated as a c = 0 CFT. Moreover this class of CFTs is important for the study of percolation or quenched disorder models in general. Sincec<sub style="">(3,2)</sub> = 0 as a minimal model only consists of the identity field, following Cardy [[http://arxiv.org/abs/condmat/0111031 9]] different approaches to get a non trivial CFT whose partition functions differ from one as suggested by the work of Pearce and Rittenberg [[http://arxiv.org/abs/mathph/0209017 69]] are presented. Concentrating on a similar ansatz for logarithmic behavior as for the triplet series (c<sub style="">(p,1)</sub>), we examine the properties of such a CFT based on the extended Kac-table for c<sub style="">(9,6)</sub>= 0 using a general ansatz for the stress energy tensor residing in a Jordan cell of rank two. We will derive the interesting OPEs in this setup (i.e. of the stress energy tensor and its logarithmic partner) and illustrate it by a bosonic field realization. We will give a motivation why the augmented minimal model seems to be more promising than the previous approaches and present an example of a tensor construction as a fourth ansatz to solve the c → 0 problem as well.  ::   [http://www.itp.uni-hannover.de/%7Eflohr/papers/anne-thesis.pdf pdf] <br><code style="font: 110% Monaco,Courier,monospace;">Journal:</code> Diploma Thesis (November 2005)
 
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Proposal for a CFT interpretation of Watts' differential equation for percolation<br><code style="font: 110% Monaco,Courier,monospace;">Authors:</code> Michael Flohr, Annekathrin Müller-Lohmann <br><code style="font: 110% Monaco,Courier,monospace;">Sources:</code>[http://www.itp.uni-hannover.de/%7Eflohr/papers.html# ]AbstractG.M.T. Watts derived in his paper[http://arxiv.org/abs/condmat/9603167 arXiv:cond-mat/9603167]that in two dimensional critical percolation the crossing probability Π<sub style="">h v</sub>satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Π<sub style="">h</sub>, Π<sub style="">h v</sub>.<br>     We will show that this differential equation can be derived from a level three null vector condition of a rational c = -24 CFT and suggest a new interpretation of the generally known CFT properties of percolation.  ::  [http://www.itp.uni-hannover.de/%7Eflohr/papers/w-percolation.pdf pdf]   ::  [http://arxiv.org/abs/hep-th/0507211 arXiv:hep-th/0507211] <br><code style="font: 110% Monaco,Courier,monospace;">Journal:</code>[http://www.iop.org/EJ/abstract/1742-5468/2005/12/P12004 J. Stat. Mech. 0512:P004 (2005)]
 
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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%;">articles</h5>
 
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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/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://dx.doi.org/10.1016/j.jcta.2006.06.010 Integrals, partitions and MacMahon's Theorem]<br>
 
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**  George Andrewsa,  Dan Romik, 2007<br>
 
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*  Slow convergence<br>
 
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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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**  George E. Andrews, 2005<br>
 
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* [http://research.microsoft.com/en-us/um/people/holroyd/papers/int.pdf Integrals, Partitions, and Cellular Automata]<br>
 
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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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* [http://www.springerlink.com/content/g420hc5h6qu6e65x/ sharp metastability threshold for two-dimensional bootstrap percolation]<br>
 
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**  Alexander E. Holroyd, 2003<br>
 
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*  Critical exponents for two-dimensional percolation<br>
 
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==relevance to dedekind eta function==
  
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*  구글 블로그 검색<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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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?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%;">experts on the field</h5>
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==related items==
  
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* [http://www.math.ubc.ca/%7Eholroyd/ http://www.math.ubc.ca/~holroyd/]
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* [[asymptotic analysis of basic hypergeometric series]]
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* http://arxiv.org/
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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%;">TeX </h5>
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* [http://arxiv.org/abs/1002.3881 A sharper threshold for bootstrap percolation in two dimensions]
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** [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]
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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]
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**  George Andrewsa,  Dan Romik, 2007
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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]
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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]
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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]
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**  Alexander E. Holroyd, 2003
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[[분류:integrable systems]]
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[[분류:math and physics]]
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[[분류:migrate]]
  
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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\(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)



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  • [{'LOWER': 'bootstrap'}, {'LEMMA': 'percolation'}]
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