Bootstrap percolation

introduction

• an 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

• growth rule
  
• http://www.math.ubc.ca/~holroyd/boot/
  

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

 

 

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
  

 

 

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

• asymptotic analysis of basic hypergeometric series
  
• Ramanujan's mock theta functions
  

 

 

articles

• A sharper threshold for bootstrap percolation in two dimensions
  
  • [1]Janko Gravner, Alexander E. Holroyd, Robert Morris, 2010
• Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation
  
  • Kathrin Bringmann, Karl Mahlburg, 2010
    
• Integrals, partitions and MacMahon's Theorem
  
  • George Andrewsa,  Dan Romik, 2007
    
• Slow convergence
  
• Partitions with short sequences and mock theta functions
  
  • George E. Andrews, 2005
    
• [HLR04]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
    
  • http://www.math.ubc.ca/~holroyd/