Solving Abrikosov's kinetic equation for the percolation cluster-size scaling function: self-consistent determination of $\sigma$ and $f(0)$, the $d=2,3$ exponent test, and the effective-perimeter kernel

Near the percolation threshold the cluster-size distribution takes the scaling form $a(n)=n^{-\tau}f(\epsilon n^\sigma)$, but the scaling function $f$ is not determined by scaling arguments alone. Abrikosov (1979) proposed a kinetic (Smoluchowski-type coagulation) equation in the threshold distance $\epsilon$ with merging kernel $W(n_1,n_2)=n_1^q n_2^q$ ($q$ an effective-surface exponent) plus an absorption term into the infinite cluster, which under the scaling ansatz reduces to a closed nonlinear integro-differential equation for $f(x)$ with exponent relations $\sigma+\tau=2q+1$ and $\delta=q/\sigma-1$ and a set of integral constraints. For the generic, physically relevant case $q<1$ (from the known critical exponents one extracts $q\approx0.73$ in $d=2$ and $q\approx0.83$ in $d=3$) the source left this equation unsolved: it states that a numerical solution is required, poses the reproduction of the accepted values of $\sigma$ in $d=2$ and $d=3$ as the decisive test of the approach, leaves the effective perimeter and $q$ without an operational definition (suggesting kernel generalizations $n^q\to n^q\phi(x)$ in a note added in proof), and leaves the predicted stretched-exponential tail on the side with an infinite cluster, $f(-|x|)\sim\exp(-t_1|x|^{q/\sigma})$, in tension with rigorous theorems and numerical evidence suggesting $\exp(-\lambda n^{(d-1)/d})$. The problem — solve the equation with its constraints, determine $\sigma$ and $f(0)$, test against $d=2,3$, and resolve the kernel-definition and tail questions — remains open.

Background

In the percolation problem, each site (or bond) of a lattice is occupied independently with probability $p$, and occupied sites joined by nearest-neighbour paths form clusters. As $p$ increases past a threshold value $p_c$ an infinite cluster appears; this percolation transition is a second-order phase transition whose order parameter is the probability $P(\epsilon)$ that a given site belongs to the infinite cluster, and whose susceptibility analogue is the mean size $R(\epsilon)$ of the finite cluster containing a given site. Central to both is the cluster-size distribution $a(n)$, the number of clusters of $n$ particles per lattice site. Near the threshold, with $\epsilon=(p_c-p)/p_c$ (so $\epsilon>0$ means $p<p_c$ and $\epsilon<0$ means $p>p_c$), scaling theory asserts the form $a(n)=n^{-\tau}f(\epsilon n^\sigma)$, where $\tau$ is the Fisher exponent, $\sigma$ the cutoff exponent, and $f$ a universal scaling function; the standard relations $\beta=(\tau-2)/\sigma$ and $\gamma=(3-\tau)/\sigma$ connect them to $P\sim|\epsilon|^\beta$ and $R\sim|\epsilon|^{-\gamma}$. Scaling arguments constrain the exponents but not $f$ itself: in two dimensions $\tau=187/91$ and $\sigma=36/91$ are known exactly, in three dimensions to high numerical precision, and on a Bethe lattice (an infinite regular tree, the mean-field case) $\tau=5/2$ and $\sigma=1/2$; the full function $f(x)$, in particular its exponential tails, is not known in closed form in any finite dimension.

Abrikosov (1979) proposed to obtain $f$ dynamically. Treating the approach to the threshold as a fictitious merging process (as the connection radius, or the occupation probability, grows, clusters coalesce pairwise), he wrote for $p<p_c$ a coagulation equation of Smoluchowski type — an integro-differential balance equation, introduced originally for aggregating colloids, in which the merger rate of clusters of sizes $n_1$ and $n_2$ is proportional to a kernel $W(n_1,n_2)$ — equating $-\partial a(n)/\partial\epsilon$ to a gain term (mergers $m+(n-m)\to n$) minus a loss term. The kernel was taken as $W(n_1,n_2)=n_1^q n_2^q$: not all particles of a cluster can participate in a merger, only those on an effective surface or perimeter, whose count scales as $n^q$ with $(d-1)/d<q<1$, interpolating between edge-only merging in one dimension ($q=0$) and full effectiveness on the Bethe lattice ($q=1$). For $p>p_c$, where finite clusters can also be absorbed into the infinite cluster, the equation acquires a loss term $-n^q a(n)S(\epsilon)$ with $S(\epsilon)=\zeta|\epsilon|^\delta\theta(-\epsilon)$ ($\zeta$ a constant, $\theta$ the Heaviside step function), interpreted as the effective perimeter density of the infinite cluster; combined with particle conservation, $\sum_{n\ge1}n\,a(n)=1-P(\epsilon)$, this also links the growth of $P$ to a $(q+1)$-moment of $a(n)$.

Substituting the scaling form into the kinetic equation yields the exponent relations $\sigma+\tau=2q+1$ and $\delta=q/\sigma-1$ (so $q>\sigma$), and a closed nonlinear integro-differential equation for $f(x)$ alone. In the regime dominated by large clusters the discrete sums are replaced by integrals, and an effective-perimeter conservation law, $\sum_n n^q[a(n,\epsilon)-a(n,0)]=-S(\epsilon)$, removes the explicit non-analytic $S(\epsilon)$ term; the conservation laws then impose integral constraints on $f$ whose convergence requires $q/2<\sigma<q$ and $\sigma>q-1/2$. Assuming $f$ analytic at $x=0$ (finite clusters at fixed $n$ should not feel the threshold singularity) gives a recursion for the Taylor coefficients of $f$ whose solvability restricts $\tau-q$ to the interval $(1,3/2)$. Asymptotically the equation predicts $f(x)\simeq c\,x^{1/\sigma-1}\exp(-t\,x^{1/\sigma})$ for $x\to+\infty$, i.e. a pure exponential $\exp(-t\,\epsilon^{1/\sigma}n)$ in $n$ at fixed $p<p_c$, consistent with known exponential decay below threshold; but for $x\to-\infty$ it predicts $f(-|x|)\sim\exp(-t_1|x|^{q/\sigma})$, a stretched exponential $\exp(-t_1|\epsilon|^{q/\sigma}n^q)$ in $n$ on the $p>p_c$ side. The scheme is exact in two limiting cases: the Bethe lattice in the Poisson limit ($q=1$, $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$, $\tau=5/2$, $\sigma=1/2$) and one-dimensional interval percolation ($q=0$, geometric distribution, though with a control-parameter-dependent rate prefactor). From the accepted critical exponents one infers $q=(1+2\beta+\gamma)/[2(\beta+\gamma)]$, giving $q\approx0.73$ in $d=2$ and $q\approx0.83$ in $d=3$.

The kinetic equation's $p>p_c$ tail conflicts with the rigorous asymptotic results on the cluster-size distribution of Kunz and Souillard (1978) and with numerical simulations in two and three dimensions, which indicate a slower decay roughly of the form $\exp(-\lambda n^{(d-1)/d})$ — the signature of compact droplets whose surface grows as $n^{(d-1)/d}$ — note that $(d-1)/d<q$. Abrikosov argued that those theorems may not apply sufficiently close to the threshold, where the infinite cluster is ramified, but did not resolve the point; in a note added in proof he observed that generalizing the kernel $n^q\to n^q\phi(x)$ with $\phi(0)=1$ would modify the $p>p_c$ asymptotic decay and could reconcile it with an asymptotic behaviour of $\ln f$ on the scale of $n^{(d-1)/d}$, and left this open. The extent to which the large-$n$ (sum-to-integral) and effective-perimeter-conservation approximations are quantitatively valid near the threshold, especially on the $p>p_c$ side and in edge cases such as $q=1$ where small-$n$ contributions matter, was likewise left as a stated limitation.

Problem Statement

The problem posed by the source, with its scope unchanged, is to complete and adjudicate Abrikosov's kinetic-equation program for the percolation cluster-size distribution.

For a fixed effective-surface exponent $q$ with $(d-1)/d<q<1$ — in particular the values $q\approx0.73$ for $d=2$ and $q\approx0.83$ for $d=3$ implied by the accepted $\beta,\gamma$ — consider the closed nonlinear integro-differential equation for the scaling function $f(x)$ obtained by inserting $a(n)=n^{-\tau}f(\epsilon n^\sigma)$, $\epsilon=(p_c-p)/p_c$, into the kinetic equation with kernel $W(n_1,n_2)=n_1^q n_2^q$ and absorption term $-n^q a(n)\,\zeta|\epsilon|^\delta\theta(-\epsilon)$, $\delta=q/\sigma-1$:

$$-\frac{\partial f}{\partial x}=\int_{0}^{1/2} u^{q-\tau} f(xu^{\sigma})\left\{(1-u)^{q-\tau} f[x(1-u)^{\sigma}]-f(x)\right\}\mathrm{d}u-f(x)\int_{1/2}^{\infty} u^{q-\tau} f(xu^{\sigma})\,\mathrm{d}u-\zeta\,\theta(-x)\,f(x)\,|x|^{\delta},$$

Here $u$ is the rescaled size of one merger partner, $u=m/n$ for a merger $m+(n-m)\to n$: the integral over $0<u<1/2$ combines the gain from such mergers with the loss of an $n$-cluster to clusters of less than half its size, the integral over $u>1/2$ is the loss to merger partners larger than $n/2$, and the last term is absorption into the infinite cluster, which acts only for $x<0$ (i.e. $p>p_c$). The system is supplemented by the equivalent large-$n$ form that uses the effective-perimeter conservation law, by the integral constraints implied by particle conservation $\sum_{n\ge1}n\,a(n)=1-P(\epsilon)$ and perimeter conservation $\sum_{n\ge1}n^q[a(n,\epsilon)-a(n,0)]=-\zeta|\epsilon|^\delta\theta(-\epsilon)$ (with convergence window $q/2<\sigma<q$, $\sigma>q-1/2$), and by the analyticity-at-$x=0$ coefficient recursion (restricting $1<\tau-q<3/2$). The open questions are:

(1) Solve this system for $f(x)$ — the source states that the generic case $q<1$ requires numerical solution — and determine, self-consistently with the constraints, the exponent $\sigma$ and the amplitude $f(0)$ for fixed $q$.

(2) Test the approach: does the resulting $\sigma$ reproduce the accepted percolation values in $d=2$ ($\sigma=36/91$) and in $d=3$ (where the accepted value is $\sigma\approx0.45$)? This is the source's explicitly posed criterion of adequacy for the kinetic approach.

(3) Clarify the physical meaning of the effective surface/perimeter and give it — and hence $q$, and possible additional $x$-dependent factors in the kernel, $n^q\to n^q\phi(x)$ with $\phi(0)=1$ — an operational or microscopic definition; in particular, determine whether such factors reconcile the predicted $p>p_c$ tail $f(-|x|)\sim\exp(-t_1|x|^{q/\sigma})$ with the slower $\exp(-\lambda n^{(d-1)/d})$ decay indicated by rigorous theorems and numerical work.

(4) Establish the conditions under which the approximations used are quantitatively valid near the threshold — in particular on the $p>p_c$ side, and in edge cases such as $q=1$ where small-$n$ contributions matter: the replacement of discrete sums by integrals (large-$n$ dominance) and the effective-perimeter conservation law.

References

1. A. A. Abrikosov, "'Kinetic equation' in the percolation theory", Journal of Physics C: Solid State Physics 12 (17), 3419-3432 (1979). DOI: 10.1088/0022-3719/12/17/015. URL: https://doi.org/10.1088/0022-3719/12/17/015.
2. H. Kunz and B. Souillard, "Essential singularity in percolation problems and asymptotic behavior of cluster size distribution", Journal of Statistical Physics 19 (1), 77-106 (1978). DOI: 10.1007/BF01020335. URL: https://doi.org/10.1007/BF01020335.
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