Percolation analysis

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

Sites, bonds, and clusters

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

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 are expected to be more realistic to capture the physics of a number of real systems 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 $x$, with $0<x<1$.
• Two sites are considered to be bonded if and only if:
  • They are nearest neighbours and
  • Both sites are occupied.

Fraction of percolating realizations

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, $X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)$, of realizations with percolating clusters. For low values of $x,(x\to 0)$ one will have $X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)\approx 0$, whereas when $x\to 1$, then $X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)\approx 1$. Considering the behavior of $X_{\mathrm{p}\mathrm{e}\mathrm{r}}$ as a function of $x$, for different values of $L$ the transition between $X_{\mathrm{p}\mathrm{e}\mathrm{r}}\approx 0$ and $X_{\mathrm{p}\mathrm{e}\mathrm{r}}\approx 1$ 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

Considering the functions $X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)$ the percolation theory predicts for large system sizes:

• $\underset{L\to \mathrm{\infty }}{lim}{X}_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)=0;\mathrm{f}\mathrm{o}\mathrm{r}x<x_c$
• $\underset{L\to \mathrm{\infty }}{lim}{X}_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)=1;\mathrm{f}\mathrm{o}\mathrm{r}x>x_c$

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

• $X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x_c,L)\approx X_{\mathrm{p}\mathrm{e}\mathrm{r}}^{(c)}$ ; for large values of $L$.

Computation of the percolation threshold

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. More sophisticated methods can be found in the literature (See Refs. for details).

Crossing of the $X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x,L)$ for different system sizes

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

Computation of pseudo-critical parameters $x_c(L)$ and extrapolation

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

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

$X_{\mathrm{p}\mathrm{e}\mathrm{r}}(x_c^{(L)},L)=1/2$.

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

$x_c(L)=x_c(\mathrm{\infty })+a{L}^{-b}$

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

Percolation threshold and critical thermodynamic transitions

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 . 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).