Bootstrap percolation

introduction

 

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

• http://mathworld.wolfram.com/BootstrapPercolation.html
  

 

 

notions of partitions

• partitions without k-gaps
  

 

 

 

tricky integrals

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

 

 

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

\(P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})\)

 

 

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 percolationAuthors: 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 Π_{h v}satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Π_h, Π_{h 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)

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• asymptotic analysis of basic hypergeometric series
  

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

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articles

• Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation
  
  • Kathrin Bringmann, Karl Mahlburg, 2010
    
• Slow convergence
  
• Integrals, Partitions, and Cellular Automata
  
  • A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368
    
• sharp metastability threshold for two-dimensional bootstrap percolation
  
  • Alexander E. Holroyd, 2003
    

• 논문정리
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html[3]
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• 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=
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

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experts on the field

• http://www.math.ubc.ca/~holroyd/
• http://arxiv.org/

 

 

TeX 

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