Bootstrap percolation
introduction
- calculation of power-law exponent for boostrap percolation
- growth rule
tricky integrals
- Henrik Eriksson: A Tricky Integral
- \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)
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)
history
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
- 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
- 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
- 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[2]
- 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)
blogs
experts on the field