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Algorithms for estimating the percolation probabilities and cluster size distribution are given in the framework of a Monte Carlo simulation for disordered lattices for the generalized site-bond problem. The site-bond approach is useful when a percolation process cannot be exclusively described in the context of pure site or pure bond percolation. An extended multiple labeling technique (ECMLT) is introduced for the generalized problem. The ECMLT is applied to the site-bond percolation problem for square and triangular lattices. Numerical data are given for lattices containing up to 16 million sites. An application to polymer gelation is suggested.

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Both computer simulations and laboratory experiments on binary reactions of random~alkers on fractal spaces bear out a recent conjecture: The time development of the reaction is dominated by the intrinsic fractal (fracton, spectral) dimension. For the Sierpinski gasket the effective spectral dimension for reactions is d, '=1. 38 (actual spectral dimension d, =1.365). For the percolating cluster (60%, square lattice) d, =1.34 (d, =1.333). From the exciton percolation laboratory experiments d, =1. 5, based on triplettrlplet annihllatlon ln naphthalene isotopic mixed crystals at 2 K.(2) where p is the density of walkers.Howcvcr, II1 mlcroscoplcally llctcrogcllcoUs IllcdIR (wc II1cludc low-dimensional spaces as special representatives of 1Mtcfogc11coUs tl1rcc-dlIlMflsloIlal spaces), onc 111ay gct dp dt txt p which is equivalent to a time-dependent rate coefficient. 7 We have conjectured' that for fractal (self-similar) spaces h is related to an effective spectral (fracton) dimensions d, ' (we usc the prime to distinguish this reaction exponent from the conventional d, ): h = 1d, '/2 (4) It follows froIIl Eq. (3) t11at tlM IIltcgratcd rate cqUat1011 Is (5) Our Monte Carlo simulations9 are for the reaction (6) No "traps" are present in these systems and all walkers Within the last year, intrinsic dynamical properties of fractals, e.g. , random walk, conductivity, phonon, and magnon effects, have been related to an unexpected new dimension dg -tlM spectral (ffRctofl) dllllcllsloI1.For EUclldcall (three-dimensional) spaces the classical behavior is retained, giving the classical diffusion constant, the Debye (T ) heat capacity at low temperatures, and Ohm's law. For fractal spaces the spectral dimension leads to drastic deviations from the classical laws. It has also been conjectured that the binary reaction coefficient is related to the same spectral dimension to which the magnon and phonon densities of states are related. 5 Our simulations bear this out for both a deterministic fractal (Sierpinski gasket) and a random one (percolating cluster).Our fusion experiments on naphthalene isotopic mixed crystals exhibit the same nonclassical bchavlor, Rt Rnd below the empirical pcrcolatlon coA" centrat1on. "Wc discuss binary reactions with one kind of reactant:where thc A species are random walkers. The standard, classical, second-order rate law gives move independently (one at a time). When two walkers occupy the same site within one instant, one walker is removed from the system and the other~alker continues its walk unperturbed.This is the only type of~alker interaction allo~ed. The planar Sierpinski gasket was generated iteratively using a method similar to Pascal's triangle. An eighth-order Sierpinski gasket was used consisting of approximately 10000 sites where we defined a site as the vertex of a triangle. If a walker is at one of the three corners of the largest trlanglc lt rcma1ns confined to thc flnltc gasket, RAd 1ts subsequent move is restricted to only one of its two possible nearest-neighbor sites. At time t=0...

Goeat-Gueat Triplet-Triplet annihilation in highly porified mixed oryntala of naphthalene (goent)naphthalene-d 8(hoat) waa atodied at 1.8 K via apectrally and time-reaolved phonphorencence and delayedflooreacence. We find nignificant deviationa from ourrent diffuaion theoriea and homogeneooa kinetion hot connintency with local heterogeneitiea (clonterization) and dynamic exciton percolation, an well nn with long-range annihilation (via noperexchnnge) There in negative evidence for nn Andernon trannition.