Bootstrap percolation

수학노트
http://bomber0.myid.net/ (토론)님의 2010년 3월 15일 (월) 14:39 판
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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
    \(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)

 

 

Conformal Field Theory and Percolation 
Authors: Annekathrin Müller-Lohmann 
Sources: [1]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 [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)

 

 

 

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