Bootstrap percolation
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
- growth rule
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
- generating function for partitions without k-gaps
- (*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,
\(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})\)
more generally, \(q=\exp(\frac{2\pi ih}{k})e^{-t}\) and \(t\to 0\) implies
\(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)
Lectures on Conformal Invariance and Percolation Lectures delivered at Chuo University, Tokyo, March 2001. pdf version
Conformal Field Theory and Percolation Authors:
Annekathrin Müller-Lohmann Sources:
[1]AbstractIn 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)
Proposal for a CFT interpretation of Watts' differential equation for percolationAuthors:
Michael Flohr, Annekathrin Müller-Lohmann Sources:
[2]AbstractG.M.T. Watts derived in his paperarXiv:cond-mat/9603167that in two dimensional critical percolation the crossing probability Πh vsatisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Πh, Πh v.
We will show that this differential equation can be derived from a level three null vector condition of a rational c = -24 CFT and suggest a new interpretation of the generally known CFT properties of percolation. :: pdf :: arXiv:hep-th/0507211 Journal:
J. Stat. Mech. 0512:P004 (2005)
history
encyclopedia
- http://ko.wikipedia.org/wiki/
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- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
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- 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 MacMahon's Theorem
- George Andrewsa, Dan Romik, 2007
- George Andrewsa, Dan Romik, 2007
- Slow convergence
- Partitions with short sequences and mock theta functions
- George E. Andrews, 2005
- 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
- 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
- Critical exponents for two-dimensional percolation
- Stanislav Smirnov, Wendelin Werner, 2001
- Stanislav Smirnov, Wendelin Werner, 2001
- 논문정리
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