"Bootstrap percolation"의 두 판 사이의 차이

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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>
 
*  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="color: rgb(180, 240, 180); font: normal normal normal 110%/normal Monaco, Courier, monospace;">Authors:</code> Annekathrin Müller-Lohmann <br><code style="color: rgb(180, 240, 180); font: normal normal normal 110%/normal Monaco, Courier, monospace;">Sources:</code> [http://www.itp.uni-hannover.de/~flohr/papers.html# ]Abstract In 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, <em>c</em><sub>(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 <em>c</em> = 0 CFT. Moreover this class of CFTs is important for the study of percolation or quenched disorder models in general. Since<em>c</em><sub>(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 (<em>c</em><sub>(<em>p</em>,1)</sub>), we examine the properties of such a CFT based on the extended Kac-table for <em>c</em><sub>(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 <em>c</em> → 0 problem as well.  ::   [http://www.itp.uni-hannover.de/~flohr/papers/anne-thesis.pdf pdf] <br><code style="color: rgb(180, 240, 180); font: normal normal normal 110%/normal Monaco, Courier, monospace;">Journal:</code> Diploma Thesis (November 2005)
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'''<em>Conformal Field Theory and Percolation </em>'''<em>'''<br>'''</em><code style="font: normal normal normal 110%/normal Monaco, Courier, monospace;"><em>'''Authors:'''</em></code><em>''' Annekathrin Müller-Lohmann <br>'''</em><code style="font: normal normal normal 110%/normal Monaco, Courier, monospace;"><em>'''Sources:'''</em></code>[http://www.itp.uni-hannover.de/~flohr/papers.html# ]<em>'''Abstract'''</em><em>'''In 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 ['''</em>[http://arxiv.org/abs/condmat/9603167 78]<em>'''] describing the horizontal vertical crossing probability within this setup. A unique solution among the minimal models, c'''</em><sub><em>'''(6,1)'''</em></sub><em>'''= -24, seems to be a good candidate, satisfying the level two differential equation for the horizontal crossing probability derived by Cardy ['''</em>[http://arxiv.org/abs/hepth/9111026 7]<em>'''] 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'''</em><sub><em>'''(3,2)'''</em></sub><em>''' = 0 as a minimal model only consists of the identity field, following Cardy ['''</em>[http://arxiv.org/abs/condmat/0111031 9]<em>'''] different approaches to get a non trivial CFT whose partition functions differ from one as suggested by the work of Pearce and Rittenberg ['''</em>[http://arxiv.org/abs/mathph/0209017 69]<em>'''] are presented. Concentrating on a similar ansatz for logarithmic behavior as for the triplet series (c'''</em><sub><em>'''(p,1)'''</em></sub><em>'''), we examine the properties of such a CFT based on the extended Kac-table for c'''</em><sub><em>'''(9,6)'''</em></sub><em>'''= 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.'''</em><em>'''  ::   '''</em>[http://www.itp.uni-hannover.de/~flohr/papers/anne-thesis.pdf pdf]<em>''' <br>'''</em><code style="font: normal normal normal 110%/normal Monaco, Courier, monospace;"><em>'''Journal:'''</em></code><em>''' Diploma Thesis (November 2005)'''</em>
  
 
 
 
 
  
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'''<em>Proposal for a CFT interpretation of Watts' differential equation for percolation</em>'''<em>'''<br>'''</em><code style="font: normal normal normal 110%/normal Monaco, Courier, monospace;"><em>'''Authors:'''</em></code><em>''' Michael Flohr, Annekathrin Müller-Lohmann <br>'''</em><code style="font: normal normal normal 110%/normal Monaco, Courier, monospace;"><em>'''Sources:'''</em></code>[http://www.itp.uni-hannover.de/~flohr/papers.html# ]<em>'''Abstract'''</em><em>'''G.M.T. Watts derived in his paper'''</em>[http://arxiv.org/abs/condmat/9603167 arXiv:cond-mat/9603167]<em>'''that in two dimensional critical percolation the crossing probability Π'''</em><sub><em>'''h v'''</em></sub><em>'''satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Π'''</em><sub><em>'''h'''</em></sub><em>''', Π'''</em><sub><em>'''h v'''</em></sub><em>'''.<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.'''</em><em>'''  ::  '''</em>[http://www.itp.uni-hannover.de/~flohr/papers/w-percolation.pdf pdf]<em>'''   ::  '''</em>[http://arxiv.org/abs/hep-th/0507211 arXiv:hep-th/0507211]<em>''' <br>'''</em><code style="font: normal normal normal 110%/normal Monaco, Courier, monospace;"><em>'''Journal:'''</em></code>[http://www.iop.org/EJ/abstract/1742-5468/2005/12/P12004 J. Stat. Mech. 0512:P004 (2005)]
  
 
 
 
 
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<h5 style="line-height: 3.428em; 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%;">articles</h5>
 
<h5 style="line-height: 3.428em; 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%;">articles</h5>
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* [http://arxiv.org/abs/1001.1977 Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation]<br>
 
* [http://arxiv.org/abs/1001.1977 Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation]<br>
 
** [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>
 
** [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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*  Slow convergence<br>
 
* [http://research.microsoft.com/en-us/um/people/holroyd/papers/int.pdf Integrals, Partitions, and Cellular Automata]<br>
 
* [http://research.microsoft.com/en-us/um/people/holroyd/papers/int.pdf Integrals, Partitions, and Cellular Automata]<br>
 
**  A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368<br>
 
**  A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368<br>
 
* [http://www.springerlink.com/content/g420hc5h6qu6e65x/ sharp metastability threshold for two-dimensional bootstrap percolation]<br>
 
* [http://www.springerlink.com/content/g420hc5h6qu6e65x/ sharp metastability threshold for two-dimensional bootstrap percolation]<br>
 
**  Alexander E. Holroyd, 2003<br>
 
**  Alexander E. Holroyd, 2003<br>
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*  Holroyd, Liggett and Romik<br>
 
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* [[2010년 books and articles|논문정리]]
 
* [[2010년 books and articles|논문정리]]

2010년 3월 15일 (월) 14:46 판

introduction

 

  • calculation of power-law exponent for boostrap percolation
  • growth rule

 

 

 

tricky integrals
  • \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)

 

 

q-series
  • definition
    \(P_k(q)\)
  • asymptotics of P_2(q)
    \(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,
    \(P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})\)

 

 

 

Andrews' conjecture

 

 

relevance to dedekind eta function

 

  • Dedekind eta function (데데킨트 에타함수)
    \(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,
    \(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})\)
    more generally, \(q=\exp(\frac{2\pi ih}{k})e^{-t}\)  and  \(t\to 0\) implies
    \(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)

 

Lectures on Conformal Invariance and Percolation Lectures delivered at Chuo University, Tokyo, March 2001. pdf version

 

 

Conformal Field Theory and Percolation 
Authors: Annekathrin Müller-Lohmann 
Sources:[1]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 [78] describing the horizontal vertical crossing probability within this setup. A unique solution among the minimal models, c(6,1)= -24, seems to be a good candidate, satisfying the level two differential equation for the horizontal crossing probability derived by Cardy [7] as well.
    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(3,2) = 0 as a minimal model only consists of the identity field, following Cardy [9] different approaches to get a non trivial CFT whose partition functions differ from one as suggested by the work of Pearce and Rittenberg [69] are presented. Concentrating on a similar ansatz for logarithmic behavior as for the triplet series (c(p,1)), we examine the properties of such a CFT based on the extended Kac-table for c(9,6)= 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.  ::   pdf 
Journal: Diploma Thesis (November 2005)

 

Proposal for a CFT interpretation of Watts' differential equation for percolation
Authors: Michael Flohr, Annekathrin Müller-Lohmann 
Sources:[2]AbstractG.M.T. Watts derived in his paperarXiv:cond-mat/9603167that in two dimensional critical percolation the crossing probability Πh vsatisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Πh, Πh v.
    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.  ::  pdf   ::  arXiv:hep-th/0507211 
Journal:J. Stat. Mech. 0512:P004 (2005)

 

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