Timeline for Understanding Finite Size Scaling in Percolation Theory
Current License: CC BY-SA 4.0
16 events
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| Jun 17, 2019 at 19:41 | comment | added | Carlo Beenakker | $F_{kj}$ are not matrix elements but expansion coefficients (the coefficient of a term $u_1^k u_0^j$), defined in equations B2 and B3 of arxiv.org/pdf/1205.1441.pdf | |
| Jun 17, 2019 at 17:22 | comment | added | user929304 | Following the notation of the cited paper, what are these matrix elements $F_{kj},$ I mean, how do we define them? For example in the Quantum Hall effect work, the cross elements $F_{01}$ and $F_{10}$ were set to $1$ but there is no mention of the diagonal elements $F_{00}$ and $F_{11}.$ How should we interpret these matrix elements? | |
| Jun 17, 2019 at 10:26 | comment | added | Carlo Beenakker | in the context where I worked on this problem, finite-size corrections turned out to be crucial to obtain a reliable value of the critical exponent. | |
| Jun 17, 2019 at 10:19 | comment | added | user929304 | Thx for the prompt rely, makes perfect sense! I think I am (hopefully) getting to the gist of this method :) One question about the Fig. you've attached here from the paper: The datapoints (without the fit) seem already to have a common crossing point, which yields the critical value of the control parameter, but were the finite size corrections (i.e terms with $u_1 L^y$) still needed to estimate the exponent correctly? because some works first say they remove the so defined finite size error from their data in order to then find a crossing point. Is such subtraction how one always proceeds? | |
| Jun 16, 2019 at 13:56 | comment | added | Carlo Beenakker | -1- indeed, the fit parameters do not depend on $L$ -2- cubic or not should not matter, but you do want to keep the dimensionality of the scaling right; so if you only scale in one direction and keep the transverse lengths fixed, you are doing a 1-dimensional scaling instead of a 3-dimensional scaling; critical exponents are dimensionality dependent, so you have to decide which dimension you want to study. | |
| Jun 14, 2019 at 14:33 | comment | added | Carlo Beenakker | in Appendix B of arxiv.org/abs/1106.5514 you see an explicit listing of the fit parameters; the Table 1 from the Kaneko-Ohtsuki paper you mention only lists the informative fit parameters (critical exponent and percolation threshold), the others are only needed for the fit and not listed. | |
| Jun 14, 2019 at 14:01 | comment | added | user929304 | (...) in their reported Tab. 1 their only fit parameters seem to have been $p_q$ and $\nu.$ This is a bit perplexing, does it mean they didn't consider the $a$ coefficients as fit parameters at all? Sorry for all these naive questions, thanks for your patience again. | |
| Jun 14, 2019 at 13:10 | comment | added | Carlo Beenakker | yes, this is the way to proceed; two things to keep in mind: -1- it is essential for the fitting procedure to be reliable that the error bars are small; a common mistake is to try to maximize the system sizes at the expense of large error bars; -2- don't overfit; keep the number of fit parameters small enough that the chi-squared value per degree of freedom is close to unity. | |
| Jun 14, 2019 at 10:39 | vote | accept | user929304 | ||
| Jun 14, 2019 at 10:38 | comment | added | user929304 | Are these indeed our fit parameters for the orders of the expansions I took in this example? I hope I haven't made any gross mistakes, just trying to get a hang of these calculations (I've tried to stick to the notation of your cited paper, wonderfully available on arXiv). Thanks for any feedback. | |
| Jun 14, 2019 at 10:37 | comment | added | user929304 | (...) For simplicity, let us assume $q_i=0, q_r=1,$ then $u_1=c_0$ and $u_0=b_1|p-p_c|.$ We insert into $P$ with $n=m_k=1$ we get $P = F_{00}(1+c_0 L^y) + F_{01} (b_1|p-p_c|^{1/\nu} + c_0 L^y b_1 |p-p_c|^{1/\nu}).$ I don't know what values $F_{00}$ and $F_{01}$ should take or if they are also fit parameters, but otherwise we have $b_1, c_0, y, \nu, p_c,$ as fit parameters, so for each curve corresponding to an $L$ we fit $P$ with those $5$ parameters. | |
| Jun 14, 2019 at 10:36 | comment | added | user929304 | (...) percolation variables w.r.t which we want to find the threshold $p_c $ or $\rho_c.$ But we have to express these in terms of relevant and irrelevant variables which we assume also to be analytic in the percolation variables if I understood correctly, i.e.: $u_0(p-p_c)=\sum_{k=1}^{q_r}b_k |p-p_c|^k$ and $u_1(p-p_c)=\sum_{k=0}^{q_i}c_k |p-p_c|^k.$ To be inserted into expanded $P=\sum_{k=0}^{n} u_1^k L^{ky} \sum_{j=0}^{m_k} u_0^j L^{j/\nu} F_{kj}.$ | |
| Jun 14, 2019 at 10:36 | comment | added | user929304 | Many thanks for this very informative answer, I had not seen this correction scheme yet, and it seems to exactly tackle what I was struggling with. Though conceptually now I understand the approach, I admit the technical aspects are bit advanced for me, I hope it's ok if I ask a few follow-up questions. Let us apply the approach to percolation problems, so the mappings might be: $g\to P,$ with $P$ the percolation probability (or average fraction of points in the largest cluster), $|p-p_c|$ or $|\rho-\rho_c|$ with $p$ the occupation probability or $\rho$ the density can be (...) | |
| Jun 13, 2019 at 20:49 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 13, 2019 at 20:41 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 13, 2019 at 15:59 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |