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### $Q$ factor: A measure of competition between the topper and the average in percolation and in self-organized criticality

##### Asim Ghosh, S. S. Manna, and Bikas K. Chakrabarti

##### Phys. Rev. E **110**, 014131 – Published 19 July 2024

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#### Abstract

We define the $Q$ factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability $p$ is increased, the $Q$ factor for the system size $L$ grows systematically to its maximum value ${Q}_{\text{max}}\left(L\right)$ at a specific value ${p}_{\text{max}}\left(L\right)$ and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of ${p}_{\text{max}}$, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at ${p}_{\text{max}}$ the value of ${Q}_{\text{max}}\left(L\right)$ diverges as ${L}^{d}$ ($d$ denoting the dimension of the lattice) as the system size approaches its asymptotic limit. We further extend this idea to nonequilibrium systems such as the sandpile model of self-organized criticality. Here the $Q(\rho ,L)$ factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with $\rho $ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.

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- Received 12 February 2024
- Revised 15 May 2024
- Accepted 8 July 2024

DOI:https://doi.org/10.1103/PhysRevE.110.014131

©2024 American Physical Society

#### Physics Subject Headings (PhySH)

- Research Areas

Percolation

- Techniques

Scaling methodsSelf-organized criticalityStatistical methods

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

#### Authors & Affiliations

Asim Ghosh^{1,*}, S. S. Manna^{2,†}, and Bikas K. Chakrabarti^{3,4,‡}

^{1}Department of Physics, Raghunathpur College, Raghunathpur 723133, India^{2}B-1/16 East Enclave Housing, 02 Biswa Bangla Sarani, New Town, Kolkata 700163, India^{3}Saha Institute of Nuclear Physics, Kolkata 700064, India^{4}Economic Research Unit, Indian Statistical Institute, Kolkata 700108, India

^{*}Contact author: asimghosh066@gmail.com^{†}Contact author: subhrangshu.manna@gmail.com^{‡}Contact author: bikask.chakrabarti@saha.ac.in

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##### Issue

Vol. 110, Iss. 1 — July 2024

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#### Images

###### Figure 1

Plot of the average number of distinct clusters per lattice site $\langle n(p,L)\rangle /{L}^{d}$ against the site percolation occupation probability $p$. For each type of lattice, the data for three different system sizes are plotted and overlap completely; only the colors used for the largest lattices are visible.

###### Figure 2

(a)Plot of the average cluster size $\langle s(p,L)\rangle $ scaled by the total number ${L}^{2}$ of lattice sites against the site percolation occupation probability $p$ for the square lattice. (b)Same data as in (a)plotted against $(1-p){L}^{1/2}$, which yields nice collapse of the data.

###### Figure 3

Plot of the percolation order parameter ${\mathrm{\Omega}}_{\text{sq}}(p,L)=\langle {s}_{\text{max}}(p,L)\rangle /{L}^{2}$ against the site percolation occupation probability $p$ for the square lattice.

###### Figure 4

Plot of the average size of the largest cluster $\langle {s}_{\text{max}}(p,L)\rangle $, average size of all clusters $\langle s(p,L)\rangle $, and $Q$ factor $Q(p,L){L}^{2}$ against the site occupation probability $p$ for a system of size $L=256$ on the square lattice.

###### Figure 5

Plot of ${Q}_{\text{sq}}(p,L)$ against the percolation occupation probability $p$ for the square lattice.

###### Figure 6

Finite-size extrapolations with suitable tuning parameters yield (a)${p}_{c}=0.592\phantom{\rule{0ex}{0ex}}717$ and $1/{\nu}_{1}=0.7574$, (b)${p}_{Q}=0.592\phantom{\rule{0ex}{0ex}}419$ and $1/{\nu}_{2}=0.9145$, and (c)${p}_{\text{max}}=0.603\phantom{\rule{0ex}{0ex}}288$ and $1/{\nu}_{3}=1.0425$.

###### Figure 7

Plot of $\mathcal{R}\left[\langle {s}_{\text{max}}(p,L)\rangle \right]$, $\mathcal{R}[\langle s(p,L)\rangle ]$, and $\mathcal{R}\left[Q\right(p,L\left)\right]$ against the site occupation probability $p$ for the square lattice of size $L=256$. The first two curves meet at point 1, where their values are equal. Therefore, their ratio is unity, which corresponds to $\mathcal{R}\left[Q\right(p,L\left)\right]=1$ at point 2.

###### Figure 8

(a)Plot of the standard deviation $\sigma \left(L\right)$ for the values of ${p}_{\text{max}}\left(L\right)$ of the square lattice against the system size $L$ on a log-log scale. The estimation of slope implies $\sigma \left(L\right)\sim {L}^{-0.658}$. (b)Plot of the average values of $\langle {p}_{\text{max}}\left(L\right)\rangle $ against ${L}^{-1.0363}$ to obtain a nice straight line. Each point is marked with its error bar. The extrapolated value of ${p}_{\text{max}}=0.6033\pm 0.0002$ has been obtained.

###### Figure 9

Plot of the difference $\mathrm{\Delta}Q=[Q(p,L)-{Q}^{\prime}(p,L)]{L}^{2}$ for six different sizes of the square lattice against the site occupation probability $p$.

###### Figure 10

Plot of the $Q$ factors against the site occupation probability $p$ for two different lattices: (a)${Q}_{\text{tr}}(p,L)$ for the triangular lattice and (b)${Q}_{\text{sc}}(p,L)$ for the simple cubic lattice.

###### Figure 11

Extrapolation of $\langle {p}_{\text{max}}\left(L\right)\rangle $ values to their asymptotic limit of $L\to \infty $ gives the estimates of ${p}_{\text{max}}$: (a)0.5088 for the triangular lattice and (b)0.3448 for the simple cubic lattice.

###### Figure 12

For the BTW sandpile, the values of ${Q}_{\text{BTW}}(\rho ,L)$ plotted against the average number of sand particles $\rho $ dropped per site of a square lattice.

###### Figure 13

For the BTW sandpile, (a)the drop densities $\langle {\rho}_{\text{max}}\left(L\right)\rangle $ for the maximal $Q$ factors and (b)the average values of the maximal $Q$ factors $\langle {Q}_{\text{max}}\left(L\right)\rangle $ plotted against different negative powers of $L$. The two colors, red and blue, represent two different types of calculations. The extrapolated values in the asymptotic limit of $L\to \infty $ are consistent with each other.