Percolation analysis

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This entry focuses on the application of percolation analysis to problems in statistical mechanics. For a general discussion see Refs. [1] [2]

Sites, bonds, and clusters[edit]

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system. Using some connectivity rules it is possible to define bonds between pairs of sites. These bonds can be used to build up clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more sequences of bonds between pairs of sites. The sites of the system can belong to different types (species in the chemistry language). Bonds are usually permitted only between near sites.

Lattice and continuum (off-lattice) models[edit]

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models. Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more realistic than lattice models to capture the physics of a number of real systems [3] [4]

Connectivity rules[edit]

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic interaction, types of sites, etc. In addition the bonding criteria can be either deterministic or probabilistic. In statistical mechanics applications one can find different bonding criteria, for example:

  • Geometric distance: Two sites, i, j, are bonded if the distance between then satisfies:  r_{ij} < R_p . This criterium is used in the so-called site percolation models (see below).
  • Probabilistic criteria: Two sites located at a certain distance  r are bonded with a given probability  b(r) , with
 0 \le b(r) < 1 . This is the case of the so-called bond-percolation models.
  • Energetic criteria (probabilistic): As an example, in the simulation of Ising models using cluster algorithms; two sites i, j, have a bonding probability given by  b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} ; where  u_{ij} is the interaction energy between sites, and  \beta = 1/ k_BT (See cluster algorithms for details).

Percolation threshold[edit]

The sizes of the clusters of a given system depend on different control parameters: density and distribution of sites, bonding criteria (which could include the effect of temperature and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the percolation threshold of the percolation transition).

Percolation and boundary conditions[edit]

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, etc. In the particular case of considering periodic boundary conditions, a cluster realization is usually considered as percolating when, at least, one of the clusters becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that appears due to the periodic boundary conditions.

Percolation and finite-size scaling analysis[edit]

Example: Site-percolation on a square lattice[edit]

Let us consider a standard example of percolation theory, [5] a two-dimensional square lattice in which:

  • Each site of the lattice can be occupied (by one particle) or empty.
  • The probability of occupancy of each site is   \left. x \right. , with  0 < \left. x \right. < 1 .
  • Two sites are considered to be bonded if and only if:
    • They are nearest neighbours and
    • Both sites are occupied.

Fraction of percolating realizations[edit]

On such a system, it is possible to perform simulations considering different system sizes (with  L \times L sites), using periodic boundary conditions. In such simulations one can generate different system realizations for given values of  x , and compute the fraction,  X_{\rm per}(x,L) , of realizations with percolating clusters. For low values of  x, ( x \rightarrow 0 ) one will have  X_{\rm per}(x,L) \approx 0 , whereas when  x \rightarrow 1 , then  X_{\rm per}(x,L) \approx 1. Considering the behavior of  X_{\rm per} as a function of  x , for different values of  L the transition between  X_{\rm per} \approx 0 and  X_{\rm per} \approx 1 occurs more abruptly as  L increases. In addition, it is possible to compute the value of the occupancy probability  x_{c} at which the transition would take place for an infinite system (that is to say, in the thermodynamic limit).

Finite-size scaling[edit]

Considering the functions  X_{\rm per}(x,L) the percolation theory predicts for large system sizes:

  •  \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0  ; {\rm for} \; \; x < x_c
  •  \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1  ; {\rm for} \; \; x > x_c

In addition, at  x = x_c , it is expected that the fraction of percolating realizations do not depend on the system size:

  •  X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)}   ; for large values of  L .

Computation of the percolation threshold[edit]

A couple of simple procedures to estimate the percolation threshold ( x_c in the example introduced above) are described here. These procedures are similar to those used in the analysis of critical thermodynamic transitions[6]. More sophisticated methods can be found in the literature (See Refs. [5] [7] [8] for details).

Crossing of the  X_{\rm per}(x,L) for different system sizes[edit]

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter  \left. x \right. and different system sizes  \left. L \right. . The critical value  x_c is then estimated by plotting  \left. X_{\rm per}(x) \right. as a function of  \left. x \right. for several values of  L . The crossing of the curves with different values of  L provide estimates of both  x_c and  X_{\rm per}^{(c)} .

Computation of pseudo-critical parameters  x_c(L) and extrapolation[edit]

Given the results of  X_{\rm per}(x,L) for a given system size  L , a pseudo-critical size dependent variable  x_c(L)=x_c^{(L)} is computed by matching  X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}.

If the universal value  X_{per}^{(c)} value is unknown for the type of transition considered, an alternative definition for  x_c\left(L \right) can be taken, for instance:

 X_{\rm per}(x_c^{(L)},L) = 1/2.

The percolation theory predicts that the pseudo-critical values  x_c(L) will scale as:

 x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b}

where  b is a critical exponent (See Refs. [1] [2] for details). Therefore, by fitting the results of  x_c(L) it is possible to estimate the percolation transition location:  x_c = x_c ( \infty ) .

Percolation threshold and critical thermodynamic transitions[edit]

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, chemical potential) as the thermodynamic transition [9] [10] [11] . In these case cluster algorithms become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the cluster algorithms page for more details).


  1. 1.0 1.1 Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533 Cite error: Invalid <ref> tag; name "Stauffer" defined multiple times with different content
  2. 2.0 2.1 Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676 Cite error: Invalid <ref> tag; name "Torquato" defined multiple times with different content
  3. Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics 89, 6427 (1988)
  4. Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E 80, 041101 (2009)
  5. 5.0 5.1 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E 72 016126 (2005) Cite error: Invalid <ref> tag; name "deng" defined multiple times with different content
  6. David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)
  7. Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E 58, 1521 - 1527 (1998)
  8. M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E 64, 016706 (2001)
  9. Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B 67 014102 (2003)
  10. Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories", Journal of Physics A: Mathematical and Theoretical 36 pp. 4269-4281 (2002)
  11. Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices", Physical Review B 40, 5007-5014 (1989)
Related reading
  • Dietrich Stauffer and Ammon Aharony "Introduction To Percolation Theory", 2nd Edition, CRC Press (1994) ISBN 9780748402533