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Lattice percolation and tracer dynamics

Summary

  • Language: C++
  • Tested compiler Intel C++ compiler icpc version 19.0.0.117 (gcc version 4.9.0 compatibility)
  • Platform: Linux or Mac
  • Publication:

    1. Giulio Biroli, Patrick Charbonneau and Yi Hu. "Dynamics around the Site Percolation Threshold on High-Dimensional Hypercubic Lattices". Physical Review E 99, 022118 (2019).
      URL: http://doi.org/10.1103/PhysRevE.99.022118

    2. Yi Hu and Patrick Charbonneau, "Percolation thresholds on high dimensional D_n and E_8-related lattices". Physical Review E (2021).
      URL: https://arxiv.org/abs/2102.09682

Dynamics around the Site Percolation Threshold on Hypercubic Lattices

Calculating the dynamical exponents of the site percolation. The model is as the following: n-dimensional (hyper)cubic lattice, with each site is randomly assigned as occupied with the probability, or "cover fraction" p. An ant on a occupied site randomly walk through the neighboring occupied sites. At each time step, the ant randomly chooses a neighbor site (in d-dimensional hypercube, a site has 2*d neighbors), and moves onto that site if it is occupied, otherwise remains in situ. The neighboring occupied sites is named as a "cluster" in this model. Hence the ant one one cluster can only reach the sites on this cluster.

Invasion percolation for hypercubic, Dn and E8-related lattices

Calculating the site or bond percolation threshold on certain lattices through the invasion percolation algorithm. The method grow a single cluster without explicitly draw the grid. Using a set to store the lattice sites and a priority queue to maintain the parameters, the fraction of the cluster size to the cluster plus perimeter size approaches to the percolation threshold.

Lattice animal enumeration for Dn lattices

Lattice animal enumeration to solve the series expansion form of the site or bond percolation threshold on Dn lattices. Usage:

# n-dimension, v-number of vertices, e-number of edges
./lanimal n v    # for site percolation
./lbanimal n e   # for bond percolation

Compiling options

Makefile has multiple labels which correspond to different programs and implementations:

  • ant Run dynamical simulations of an ant on the lettice. There are different implementations based on different background data structure, including:

    • ant_ord The visited sites are stored in std::set, based on red-black tree, with O(log n) query time. It is adapted to the case of p>p_c where large clusters might be possible.
    • ant_und The visited sites are stored in std::unordered_set, based on hash table, with O(1) query time with more memory . It is adapted to the case of p <= p_c where large clusters is unlikely, otherwise the program may run out of the memory when a large cluster is encountered.
    • ant_hbd Solving dynamics on small clusters by transfer matrix and do simulations only on large clusters. Uses ant_und implementation as the parent class, which means the simulation part is the same as ant_und. One can easily modify the code to call ant_ord implementation if necessary.

  • msdlim Calculate the infinite time limit of MSD on a lattice

  • leath Calculate the infinite time limit of MSD by Leath algorithm, i.e., growing a cluster from the origin

  • invade Invade percolation algorithm to compute the percolation threshold. An implementation of the method of S Mertens, and C Moore. Percolation Thresholds and Fisher Exponents in Hypercubic Lattices. Phys. Rev. E 98, 022120 (2018)

    (05/2021) Updated for computing the percolation threshold of both site and bond percolation for a variety of lattices, including the hypercubic lattice, Dn lattice, E_6, E_7, E_8 and Lambda_9 lattices.

  • lamimal.make separate makefile for generating lattice animal enumeration

File list

  • AntLatticeWalk
    • ant.hpp
    • cluster_counter.hpp
    • leath_counter.hpp
    • ant_global.cpp
    • coordinate.h
    • leathsitenode.hpp
    • ant_global.hpp
    • grid_field.h
    • limited_queue.hpp
    • ant_main.cpp
    • msdlimit.cpp
    • ant_ord.cpp
    • invade.hpp
    • read_input.cpp
    • ant_und.cpp
    • leath.cpp
    • read_input.h
    • cluster_counter.cpp
    • leath_counter.cpp
  • DIM
    • *D/dim.h Headers define the dimension of systems when compiling
  • DnLatticeAnimal
    • animal.cpp
    • animal_bond.hpp
    • animal_site.hpp
    • coordinate.h
    • leathsitenode.h
    • main_perimeter.cpp
    • main_count.cpp
    • main_bondcount.cpp
  • HybridAnt
    • ant_hbd_main.cpp
    • ant_hbd.hpp
    • ant_mat.cpp
    • ant_hbd.cpp
  • InvadePercolation
    • basevec.cpp
    • invade.cpp
    • invade.hpp
  • MSDDumpReader Small data processing programs to compute the local $\mu_-$ (MSDDumpReader in d<6; MSDDumpReaderB for the prefactor in d >= 6)

Example input files

  • inputfile_ant.dat for ant_ord, ant_und, ant_hbd
  • inputfile_ivd.dat for invade percolation
  • inputfile_leath.dat for leath
  • inputfile.dat for msdlim

To compile

One may need to manually configure the environment variables INTEL_ROOT, MKL_ROOT to target the path of intel compiler and MKL librarys. Only ant_hbd need MKL_librarys; other programs should be compatible on g++, clang++ and other C++ compilers.

To compiler one of the target, for example, leath, for 3D systems, run command make leath DIM=3D

and so on.