Generating correlated disorder on hypercubic lattices

This is a general implementation of the Fourier Filter Method for generating correlated disorder on a hypercubic lattice. We provide all relevant functions in header-only c++ files that can be easily included in any existing code. An example file shows how to generate long-range power-law correlated disorder.

This implementation was used to produce the data in [1] for C(r)=(1-r^2)^{-a/2}. Consequently, long-range correlated disorder generated according to the respective function (see below) should show percolation thresholds and fractal dimensions according to [1]. However, we do not take responsibility for any deviations.

Authors: Johannes Zierenberg, Niklas Fricke, Martin Marenz, F. Paul Spitzner, Viktoria Blavatska and Wolfhard Janke

If you use parts of this code please cite [1].

Requirements

• C++11 compatible compiler (tested with GNU gcc)
• FFTW3 installed (e.g. on debian 'sudo apt-get install libfftw3 libfftw3-dev')

Usage

The header-only functions need to be included in a c++ program. The following brief manual gives a rough idea of the steps

1. Steps to generate a d-dimensional lattice of linear size L=2^l with long-range power-law correlated site variables:

   1. construct a Disorder object, e.g.

      Disorder myDisorder(d,l)  
      

   2. choose a correlation function:

      • init_correlation_power_law_euclidean(double a):

      C(r) = (1+|r|^2)^(-a/2); |r|=sqrt(\sum r_i^2)
      

      • init_correlation_power_law_manhattan(double a):

      C(r) = (1+|r|^2)^(-a/2); |r|=\sum r_i          
      

      • init_correlation_power_law_euclidean(double a, double alpha):

      C(r) = (1+|r|^{alpha})^(-a/alpha); euclidian 
      

      • init_correlation_custom(std::vector& Cx_)

      C(r) = custom
      

   3. if several disorder are required sequentially, then precompute the spectral density by calling

      myDisorder.precompute_sqrt_Sk();
      

   4. each call to generate a correlated disorder yields two independent disordered lattices. Predefine lattices

      std::vector<double> lattice1(disorder.N,0.0);
      std::vector<double> lattice2(disorder.N,0.0);
      

      And generated them

      myDisorder.generate_correlated_continuous(lattice1, lattice2, seed);
      

2. Steps to generate discrete lattices with p=density of non-defect sites:

   1. Repeat 1)
   2. Be aware that sigma2=C(0) and call

      discretize( lattice1, calc_theta_global(p,sigma2) );
      

      The lattice1 now holds the discretized lattice

Example

For an example first build via

make

and then executed

./example

The output can be compared to the data provided in

./output/

where we also provide a gnuplot file to generate the corresponding disorder illustrations. Again, we provide a Makefile to generate the figures via

make

File description

FourierTransfromWrapper.hpp

Wrapper arround fftw.

correlated_disorder_continuous.hpp

All functions needed to generated continuous variables on a hypercubic lattice correlated according to predefined and customized correlation functions.

correlated_disorder_discrete.hpp

Functions needed to map the continuous site variables to discrete ones. This also includes functions we used to measure the percolation threshold.

example_generate_configuration.cpp

Example test program which produces the same files as provided in

./output/

References

[1] J. Zierenberg, N. Fricke, M. Marenz, F.P. Spitzner, V. Blavatska, and W. Janke, "Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects", Phys. Rev. E 96, 062125 (2017).

[2] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd edition: The Art of Scientific Computing (Cambridge University Press, Cambridge, 2007).