Bootstrap percolation

introduction

 

• calculation of power-law exponent for boostrap percolation
  

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

 

• Andrews' conjecture
  

 

 

integrals

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

Henrik Eriksson: A Tricky Integral

 

 

 

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
  \(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)
  

 

 

 

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=

[[4909919|]]

 

 

articles

 

• Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation
  
  • Kathrin Bringmann, Karl Mahlburg, 2010
    
• 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
    
• Holroyd, Liggett and Romik
  

• 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[1]
• 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

• 구글 블로그 검색
  
  •  
    
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

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

 

 

TeX 

• Detexify2 - LaTeX symbol classifier