Nested Closed Paths in Two-Dimensional Percolation

Percolation BroadbentHarmmersley57 ; StaufferAharony1994 ; Grimmett1999 ; BollobasRiordan2006 is a paradigmatic model in the field of phase transitions and critical phenomena and a central topic in probability theory. It also finds important applications in various fields such as fluids in porous media hunt2014percolation , gelation stauffer1982gelation and epidemiology Meyers2007 . Bond percolation corresponds to the $Q\!\to\!1$ limiting case in the Fortuin-Kasteleyn cluster representation of the $Q$-state Potts model Potts ; FK , and provides a simple illustration of many important concepts for the latter FYWu . In two dimensions (2D), the algebraic use of symmetries—lattice duality KW41 , Yang-Baxter integrability Lieb67 ; Baxter72 and conformal invariance BPZ84 ; FQS84 —has led to a host of exact results. Critical exponents are predicted by Coulomb-gas (CG) arguments Nienhuis1987 , conformal field theory Cardy1987 and Stochastic Loewner Evolutions LawlerSchrammWerner2001 , and have been proven rigorously for e.g. triangular-lattice site percolation Smirnov2001 .

In site percolation, the sites of a lattice are occupied with probability $p$ and empty otherwise. A sequence of distinct, occupied sites of which each is nearest neighbor to its predecessor is called a path. Two occupied sites are connected if there is a path from one to the other. Two paths are independent if they do not have a site in common. A closed path has neighboring first and last sites. A maximal set of connected sites is called a cluster.

In bond percolation, the edges or bonds of the lattice are occupied with probability $p$, or vacant (empty). Paths, connectivity and clusters follow the same definitions as in site percolation, with sites replaced by bonds of the lattice, and nearest neighbor by having a site in common. Two paths are independent if they do not have a bond in common, and do not cross, but they may share a site.

Clusters and paths can also be introduced for empty elements. For bond percolation, these paths and clusters typically consist of bonds on the dual lattice. A dual bond is occupied if the (primal) bond it intersects is empty, and vice versa. For this reason, the paths on empty elements are often referred to as dual.

A path between two regions is typically not unique. In contrast, cluster boundaries form trajectories which are uniquely determined by the configuration. Two different cluster boundaries are by nature non-overlapping.

Cluster boundaries define two families of exponents, which have been computed by CG methods. The so-called watermelon exponents BN1984 ; DupSal1987 govern the probability that two (or more) distant regions are connected by a given number, say $n$, of cluster boundaries. The watermelon exponents have the value $X_{\textsc{wm}}(n)=(n^{2}-1)/12$.

The second, continuous family governs the correlation function of what we here here call the nested-loop operator (see dNijs1983 ; MitNie2004 ). This operator gives a weight, say $k$, to each cluster boundary surrounding the insertion point (multiple such boundaries must be nested, whence the name). Its two-point function gives a weight to those cluster boundaries that surround one, but not both of the insertion points. The corresponding exponent is

Analogous to $X_{\textsc{wm}}(n)$ are the so-called monochromatic $n$-path exponents $X_{\textsc{mp}}(n)$, governing the decay of probability that between two distant regions there are $n$ independent paths, all on clusters (or, equivalently, all on dual clusters). The case $n\!=\!1$ means that no cluster boundary separates the two regions, so $X_{\textsc{mp}}(1)\!=\!X_{\textsc{nl}}(0)\!=\!\frac{5}{48}$. But the exponents for $n=2$ (backbone exponent) or higher $n$ do not appear amenable to CG analysis, and hence they are known only numerically JacZinn02 ; BefNol2011 ; Xu2014 . As a side-remark we mention that when one or more, but not all, of the paths are on dual clusters, the exponents are different and in fact identical to $X_{\textsc{wm}}(n)$ Aizenman1999 ; BefNol2011 .

In this Letter we propose to similarly consider the path analogue of the nested-loop operator: the nested-path (NP) operator. It gives a weight $k$ to each independent closed path surrounding the insertion point. We investigate here the exponent of this operator by numerical means. For simplicity, we simulate its one-point function: 2D percolation with the NP operator placed at the center of a compact domain of linear dimension $L$. We thus estimate the expectation value $W_{k}\equiv\langle\mathcal{R}\cdot k^{\ell}\rangle$, where $\mathcal{R}$ is the indicator function of a path from the center to the boundary of the domain, and $\ell$ the number of independent closed paths surrounding the center. The factor $\mathcal{R}$ ensures that two consecutive surrounding paths are connected, rather than separated by two cluster boundaries. The configurations with $\mathcal{R}=1$ we call percolating.

We show that, at the percolation threshold, the scaling of $W_{k}$ obeys a power law $W_{k}\!\sim\!L^{-X_{\textsc{np}}}$ with an exponent $X_{\textsc{np}}(k)$, that depends continuously on the weight $k$. A high-precision estimate of $X_{\textsc{np}}$ is obtained as a function of $k$. For $k=2$, we observe that $W_{2}(L)=1$ for site percolation on the triangular lattice with any $L$, and prove this to be true for any self-matching lattice SykesEssam64 . We present an analytical formula (3), analogous to (1), which reproduces some exact values and agrees so well with the numerical results, that we conjecture it to be exact.

We study site percolation on a triangular lattice (STr) in a hexagonal domain with free boundaries. The scale $L$ is the length of the diagonal. Fig. 1a shows a sample configuration, with $L=17$. The central site is neutral, and the other sites are occupied with probability $p$. For each configuration we calculate the number $\ell$ of independent closed paths that surround the center, and $\mathcal{R}$ which is 1 (0) if there is (not) a path from the center to the boundary. While $\ell$ is well-defined, the paths are not unique. In Fig. 1a, $\mathcal{R}=1$ and $\ell=3$.

Analogous procedures are applied to bond percolation on the square lattice (BSq), see Fig. 1b, with a neutral bond placed at the center, and the length of the diagonal, $L=15$.

We are interested in the scaling behavior of $W_{k}(L)$ at the percolation threshold $p_{c}$. For $k=1$, $W_{1}\equiv\langle\mathcal{R}\rangle$ represents the probability that the central site is connected to the boundary, which is seen from (1) to decay as $\langle\mathcal{R}\rangle\sim L^{-5/48}$. The contributions to $W_{0}$ are those, which have a path from the center to the boundary, but no closed path surrounding the center. These configurations must have a path of occupied and one of empty elements from the center to the boundary and consequently two cluster boundaries connecting the center to the boundary. These events are selected for the watermelon exponent $X_{\textsc{wm}}(2)=1/4$. Thus, proposing $W_{k}\sim L^{-X_{\textsc{np}}(k)}$, we already know that $X_{\textsc{np}}(1)=5/48$ and $X_{\textsc{np}}(0)=1/4$.

We carry out extensive simulations for STr and BSq at the percolation threshold $p_{\rm c}=1/2$, with geometric shapes as in Fig. 1. The linear size $L$ is taken in range $3\leq L\leq 8189$. For both models and each $L$, the number of samples is at least $5\times 10^{9}$ for $L\leq 100$, $2\times 10^{8}$ for $100<L\leq 1000$, $2\times 10^{7}$ for $100<L\leq 4000$, and $4\times 10^{5}$ for $L>4000$.

A log-log plot of $W_{k}$ versus $L$ is shown in Fig. 2. The data clearly support asymptotic power-law dependence of $W_{k}$ on $L$. We fit (by least squares) the $W_{k}$ data to

We admit only data points with $L\geq L_{\rm m}$ for the fits, and systematically study the effect on the residual $\chi^{2}$ value (weighted according to confidence level) when varying $L_{\rm m}$. In the best fits, $\omega\approx 1$. The results with $\omega=1$ are given in Tab. 1. The estimates of $X_{\textsc{np}}(1)$ and $X_{\textsc{np}}(0)$ agree well with the exact values $5/48$ and $1/4$, respectively. Table 1 strongly indicates that $X_{\textsc{np}}(2)=0$ for $k=2$.

The $W_{2}$ data for STr and BSq are listed in Tab. 2 and plotted in Fig. 3 versus $L$. For STr we find for $L\leq 7$ by exact enumeration, and for larger $L$ within statistical errors, that $W_{2}=1$. For BSq, we find $W_{2}=1$ for $L=3$ and $2097075/2^{21}$ for $L=5$, shown in Tab. 2 together with simulation data for larger $L$. As $L$ increases, $W_{2}$ converges to a value slightly smaller than, and clearly different from 1. A least-squares fit $W_{2}(L)=W_{2,\infty}+bL^{-2}$ for $L>20$, gives the asymptotic value as $W_{2,\infty}=0.994\,5(2)$. Thus, we conjecture $X_{\textsc{np}}(2)=0$ in general, and, for STr, $W_{2}(L)=1$.

Then, in an attempt to find a formula analogous to (1), we set $k=2\cos(\pi\phi)$ and plot $X_{\textsc{np}}$ as a function of $\phi^{2}$, as shown in Fig. 4. Noting (i) $X_{\textsc{np}}(2)=0$, (ii) an apparent pole for some positive $\phi^{2}<1$, and (iii) an asymptotic slope of 3/4 for negative $\phi^{2}$, just as $X_{\textsc{NL}}$, (1), it is natural to propose $X_{\textsc{np}}=3\phi^{2}/4+a\phi^{2}/(\phi^{2}-b)$ as the simplest rational function that matches these observations. Then the exact results for $X_{\textsc{np}}(0)$ and $X_{\textsc{np}}(1)$ fix

in which some well-known exponents of 2D percolation seem to appear. The excellent agreement with the numerical results shown in Fig. 4 and Table 1, leads us to conjecture that (3) is exact. But, in spite of some similarity with (1) we have not found any theoretical basis for (3).

In simulations, each site (bond) is randomly occupied with probability $p_{c}=1/2$, a cluster is grown from the center by standard breadth-first search. If the cluster does not reach the boundary, ${\mathcal{R}}=0$ and there is no contribution to $W_{k}$. Otherwise, we compute the number $\ell$ of paths surrounding the center. This can be achieved efficiently by the following algorithm.

The $m$-th path surrounding the center acts as the seed of one or more dual (empty) clusters, linked together by the $(m-1)$-th surrounding path. By growing the dual clusters from the $(m-1)$-th surrounding path, one can locate the $m$-th path as the chain of occupied sites (or bonds) that stops the dual-cluster growth. A caveat is that for BSq the dual cluster consists of bonds of the dual lattice. Algorithm 1 sketches the corresponding procedure, in which $\Omega$ is the region that is encircled by the next surrounding path. It is written for STr, but when sites are replaced by bonds, and adjacent sites by bonds sharing a site, it works for BSq too.

$W_{2}(L)=1$ for STr — Take an arbitrary STr configuration on a simply connected piece of the lattice, with a distinguished, ‘central’ site. Consider a maximal set of independent paths surrounding the center, with each path consisting either of only occupied (red) sites or of only empty (green) sites. The innermost such colored paths, given the interior ones, are uniquely defined (and can be constructed by Algorithm 1).

Let $\ell$ be the number of these nested paths. By $P_{i}$ we denote the map that inverts (red $\leftrightarrow$ green) the color of path $i$ (counted from the center), and of all sites strictly between path $i$ and path $i-1$ (or between path 1 and the center). Fig. 5 shows four configurations, related to each other by the maps $P_{i}$.

Since $p_{c}=\frac{1}{2}$, the set $\{P_{i}\}_{i=1}^{\ell}$ generates $2^{\ell}$ equiprobable configurations. Within this ensemble, ${\cal R}=1$, if and only if all the paths are occupied. The entire ensemble can be generated by the $P_{i}$ from any of its members. Therefore all configurations are member of precisely one such ensemble, and as a consequence $W_{2}=\langle\mathcal{R}\cdot 2^{\ell}\rangle=1$.  $\Box$

An essential property used in the proof is that inverting a path of occupied sites surrounding the center creates a barrier preventing a path of occupied sites to connect the center with the boundary. This is an obstacle against applying the proof to BSq, where a path of occupied bonds can cross a path of empty bonds, see SupMat for more explanation.

The regularity of the lattice and of the domain are not used in the proof, but having $p_{c}=\frac{1}{2}$ is crucial. However, $p_{c}=\frac{1}{2}$ for any self-matching lattice, and in particular for lattices of which all faces are triangles. Hence, the result is also valid for regular or irregular planar triangulation graphs, of any shape and position of the center.

We construct a new family of geometric quantities $W_{k}$ for critical percolation in two dimensions and determine a continuous set of critical exponents $X_{\textsc{np}}(k)$ with high precision. An identity $W_{2}(L)=1$, independent of the linear size $L$, is found for triangular-lattice site percolation and proven for any lattice with only triangular faces. This implies an exact exponent $X_{\textsc{np}}(2)=0$. The universality of the critical exponent $X_{\textsc{np}}$ is well supported by simulations for both bond and site percolation. Apart from the special cases $k=2,1,0$, the exact values of $X_{\textsc{np}}$ are unknown. We conjecture an analytical function, Eq. (3), which reproduces the known exact results and agrees excellently with numerical estimates of $X_{\textsc{np}}$ for other $k$ values. We note that, though Eq. (3) is somewhat similar to existing results, proving it remains elusive.

Future work will involve the $Q$-state Potts model in the Fortuin-Kasteleyn cluster representation, which includes bond percolation as a special case for $Q\rightarrow 1$.