"Bootstrap percolation"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
38번째 줄: 38번째 줄:
 
*  Andrews and Zagier expression of <math>P_k(q)</math><br>
 
*  Andrews and Zagier expression of <math>P_k(q)</math><br>
 
*  result of '''[HLR04]'''<br> if <math>q=e^{-t}</math> and  <math>t\sim 0</math><br><math>P_k(q) \sim \frac{-\lambda_k}{1-q}</math> as <math>q \to 1</math><br>
 
*  result of '''[HLR04]'''<br> if <math>q=e^{-t}</math> and  <math>t\sim 0</math><br><math>P_k(q) \sim \frac{-\lambda_k}{1-q}</math> as <math>q \to 1</math><br>
−
 
−
 
 
  
 
 
 
 
66번째 줄: 64번째 줄:
 
<h5 style="line-height: 2em; margin: 0px;">relevance to dedekind eta function</h5>
 
<h5 style="line-height: 2em; margin: 0px;">relevance to dedekind eta function</h5>
  
−
*  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})</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>
  
 
 
 
 
116번째 줄: 114번째 줄:
 
<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>
 
<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>
  
 +
* [http://arxiv.org/abs/1002.3881 A sharper threshold for bootstrap percolation in two dimensions]<br>
 +
** [http://arxiv.org/abs/1002.3881 ]Janko Gravner, Alexander E. Holroyd, Robert Morris, 2010
 
* [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>

2010년 7월 28일 (수) 15:51 판

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

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books

 

[[4909919|]]

 

 

articles

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

TeX 
원본 주소 "https://wiki.mathnt.net/index.php?title=Bootstrap_percolation&oldid=37163"