{'fraction_non_alphanumeric': 0.033890436397400185, 'fraction_numerical': 0.01137418755803157, 'mean_word_length': 4.654199475065616, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Two-dimensional percolation with multiple seeds

We study non-uniform percolation in a two-dimensional cluster growth model with multiple seeds. With increasing concentration of seeds, the percolation threshold is found to increase monotonically, while the exponents for correlation length, order parameter, and average cluster size, keep invariant. The scaling law for an infinite square lattice keeps working for any nonzero concentration of seeds. Abnormal finite-size scaling behaviours happen at low concentration of seeds.


I. INTRODUCTION
In an ordinary two-dimensional percolation model, the sites or bonds of a lattice are usually distributed uniformly, whether the lattice is square, triangular, diamond, or in other forms [1]. However, in nature, the non-uniform distribution may be more popular than the uniform one. For example, electron density is used to describe the non-uniform spatial distribution of an electron in materials [2], cancer cells begins in some tissues or organs of the body, but not the whole body. In view of the wide influence of percolation theory [3][4][5][6][7][8][9], it is essential to study the nonuniform percolation models and their percolation properties.
In the past decades, an important development towards nonuniform percolation is the study on the correlated percolation, examples of which are bootstrap percolation [10][11][12][13], jamming percolation [14][15][16][17], and directed percolation [18][19][20][21][22][23][24][25] etc. Our model is certainly some kind of correlated percolation model. However, our model originates from a completely different idea. In our model, clusters start to grow from a number of preoccupied sites. Our model is at first a cluster growth model. Except one special case, the model displays similar properties as the ordinary non-restricted percolation model. The only difference is the specific values of percolation thresholds. The critical properties of the model are obtained in the same way that already used in an ordinary percolation model.
For the convenience, we summarize the formulae here, detailed description of the method can be found elsewhere [1]. An interesting quantity of a percolation model is the correlation length defined as where n s is the average number of s-clusters per lattice site, 2R 2 s = ij |r i − r j | 2 /s 2 is the average squared distance between two cluster sites. It is expected to behave as infinite system, Π = 1 above and Π = 0 below p c . As for percolation transitions, only the value of p c is not enough, we have to introduce a number of other observables. On a square lattice with periodic boundary conditions, the quantity dΠ/dp gives the probability per interval dp at concentration p, that a wrapping cluster appears for the first time. The average concentration p av , at which, a wrapping cluster appears for the first time is defined as p av = p dΠ dp dp.
If we define the width ∆ of the transition region as then ∆ can be related to p av via In this way, one can first get the value of p c by fitting the observed thresholds p av and the observed widths ∆, without prior knowledge of the correlation length exponent ν. With the value of p c at hand, one can further obtain the value of ν through If an observable X is predicted to scale as |p − p c | −λ in an infinite lattice, then we expect it to obey the general scaling law whereX is a scale-independent function. Other two interesting observables are the order parameter, P ∞ , the probability of an occupied site belongs to the infinite cluster, and the average cluster size, S. At p = p c , they behave respectively as Here, the value of P ∞ (or S) at p = p c is estimated by linear interpolation between two values of P ∞ (or S) right above and below p c . The exponents for two-dimensional lattice are expected to obey the well-known scaling law These exponents will be calculated in our model.

II. MODEL
Our model can be easily built up by first occupying a number of sites randomly chosen from a two-dimensional square lattice with N = L 2 sites, then one by one, occupy the empty sites being neighbour to the previous occupied sites. After occupying a qualified empty site (being neighbour to at least one of the previous occupied sites), the list of qualified empty sites is refreshed by adding the new empty sites being neighbour to the site that was occupied just now. Groups of neighbour sites form clusters. Thus clusters grow from the multiple seeds, the sites occupied at the very beginning. Since the later occupied sites in our model are surrounding the previously occupied sites, there could be some differences between our model and other models. We use η to denote the concentration of seeds. For the Eden model [26,27], η = 1/N, which approaches zero with increasing L. When η ≥p c , withp c the percolation threshold of the usual two-dimensional site percolation [28][29][30], the critical region of percolation transition will be covered by the occupying process of the sites chosen as seeds, the following cluster growth process is therefore less meaningful. Especially, η = 1 is the usual percolation model. So, it is of interest only in the region with 0 < η <p c .
We have calculated the thresholds p av for many sets of η and L. insensitive to the value of η as it should be. This peak value is larger than that in the freeboundary lattice with the same L, because touching-boundary clusters could become bigger by including the sites across the periodic boundaries, and a spanning cluster not counted in calculating ξ should be summed if it does not form a wrapping cluster when the freeboundary condition is switched to the periodic-boundary condition [30]. It is worth of noting that, the peak value of ξ appears in different point, one is around 0.54, the other is around 0.58, which implies that percolation thresholds for models with different concentration of seeds are different. Obviously, the p c values decrease with decreasing η values. In other words, if the number of seeds for cluster growing is smaller, the percolation phase transition will happen earlier.
This result seems a little bit strange but is understandable. In our model, the following occupied sites are gathered to the clusters centered with these seeds occupied at the very beginning. Given more seeds, which means there are more clusters growing from these seeds to randomly distribute all the occupied sites, thus the largest cluster in this case will be smaller, and the wrapping cluster will be certainly delayed to appear. Data fitting gives of exponents in an ordinary two-dimensional site percolation model. The ratio between exponents (2β + γ)/ν is therefore a constant, and it obeys the scaling law Eq. (10) as it does in an usual site percolation model.

IV. CONCLUSION
Given a number of seeds (except only one seed) on any lattice, percolation phase transition will inevitably happen while clusters centered with these seeds grow up. Except the universal scaling law and various scaling exponents, the percolation threshold is variable Here, p c = 0.530298 for the model η = 0.1. with non-uniform distribution of sites characterized by the concentration of seeds. Percolation properties of a lattice depend on the non-uniform distribution of sites or bonds centered with multiple seeds, as well as its structures. Any details (uniform or non-uniform structures) of geometrical distribution of sites or bonds may contribute to the geometrical phase transition of a lattice. In general, smaller number of seeds for clusters growing implies the earlier occurrence of percolation transition. It is expected that there exists a critical value of the concentration of seeds, only below which, the abnormal finite-size scaling behaviours could happen.
The idea of multiple seeds can be extended to other correlated percolation models. This work make it possible to push the application of percolation theory to wider fields, where percolation thresholds are expected to vary with non-uniform population of sites or bonds while scaling exponents keep invariant.
Upon finishing this work, we became aware of a similar cluster model with initial seed concentration ρ and an additional parameter called growth probability g reported by Roy and Santra recently [31]. However, our model (corresponding to their g = 1) has not been discussed there. Except the difference in the model itself, the results of their model, including the values of percolation thresholds around 0.593 for the selected seed concentration 0.05, 0.25, and 0.50, and all the exponents are nearly the same for an ordinary percolation model.