Breaking universality in random sequential adsorption on a square lattice with long-range correlated defects
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Breaking universality in random sequential adsorption on square lattice withlong-range correlated defects
Sumanta Kundu ∗ and Dipanjan Mandal † Department of Earth and Space Science, Osaka University, 560-0043 Osaka, Japan Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: February 26, 2021)Jamming and percolation transitions in the standard random sequential adsorption of particleson regular lattices are characterized by a universal set of critical exponents. The universality class ispreserved even in the presence of randomly distributed defective sites that are forbidden for particledeposition. However, using large-scale Monte Carlo simulations by depositing dimers on the squarelattice and employing finite-size scaling, we provide evidence that the system does not exhibit suchwell-known universal features when the defects have spatial long-range (power-law) correlations. Thecritical exponents ν j and ν associated with the jamming and percolation transitions respectively arefound to be non-universal for strong correlations and approach systematically to their own universalvalues as the correlation strength is decreased. More crucially, we have found a difference in thevalues of the percolation correlation length exponent ν for a small but finite density of defects withstrong spatial correlations. Furthermore, for a fixed defect density, it is found that the percolationthreshold of the system, at which the largest cluster of absorbed dimers first establishes the globalconnectivity, gets reduced with increasing the strength of the spatial correlation. I. INTRODUCTION
The study of adsorption of particles onto solid surfacesis a subject of great interest in different disciplines of sci-ence and technology [1–4] due to its relevance in diverseapplications, including protein adsorption [5], ion im-plantation in semiconductor [6], and thin film depositiontechnologies for surface coatings and encapsulations [7].In the simplest case of adsorption leading to monolayerformation, such as the binding of protein molecules onglass or metals [2], one considers that the process of ad-sorption takes place irreversibly and the particles have nomobility. Consequently, they remain at their position ofadsorption forever. However, many complex dynamicalphenomena, such as diffusion, desorption, and thermalexpansion of particles are often found to be associatedwith the process of adsorption occurring in the real-worldsystems [8–10]. It has been observed that such underly-ing mechanisms crucially affect the morphology of thegrowing monolayer formations. Apart from that, prop-erties of the surface, for example, the surface roughnessor the interfacial interaction play a significant role in thekinetics of adsorption [11]. However, to our knowledge,the later aspects have not been studied in great detailsusing theoretical models.The theoretical study of monolayer formation in thelimit of irreversible adsorption has been carried out quiteintensively over the last several decades through thestochastic models of random sequential adsorption (RSA)[1, 12, 13]. In the standard RSA, particles are absorbedsequentially and irreversibly at random positions ontoan initially empty substrate subject to a constraint that ∗ [email protected] † [email protected] they only interact through excluded volume interaction.The kinetics of adsorption terminates when a jammingstate is reached where no more vacant space is availableto accommodate a single particle. The surface coverage p , defined as the volume fraction of the surface occupiedby the adsorbed particles, attains a non-trivial value p j at the jamming limit. The exact value of p j is knownonly for one-dimensional systems in both continuum andlattice spaces [14, 15].Another important aspect which has been studied us-ing the RSA model, is the phenomenon of percolation ofpoly-atomic species [16–19]. A group of adsorbed parti-cles, occupying more than one lattice site, form clustersthrough their neighbouring connections. The percolationtransition occurs when such a cluster connects two oppo-site boundaries of the system through a spanning path ata critical value of the surface coverage p = p c , known asthe percolation threshold. Therefore, the global connec-tivity exists in the system only in the percolating phaseof p > p c . The system exhibits the generic scale-invariantfeatures of a continuous phase transition right at p c [20].Furthermore, the role played by the shape and size ofthe depositing particles on the morphology of the grow-ing structure have been studied [16, 18, 21–23]. Differ-ent mechanisms of adsorption have also been introducedto explain various experimental observations comprehen-sively [10, 24–26]. It has been revealed that the jam-ming density p j and the percolation threshold p c dependnon-trivially on all these factors. However, interestingly,the critical behavior of the system associated with thetwo transition points are found to be universal, meaningthat they are characterized by a universal set of criticalexponents that are independent of all these microscopicdetails. In all these cases, the percolation transition be-longs to the ordinary percolation universality class [17–19, 22, 25, 26]. Similarly, the jamming transition is char-acterized by the universal exponent ν j relating to the sizescaling of the width ∆ of the transition zone, which scaleswith linear size L of the substrate in a spatial dimension d as ∆ ∼ L − /ν j , with ν j = 2 /d . Robustness of this uni-versal scaling law has been examined on the Euclideanand fractal lattice geometries [27, 28].Although the effect of particle properties on the ki-netics of adsorption have been extensively studied in thepast, a theoretical investigation on the role of surfaceproperties has remained almost unexplored. In this con-text, some previous studies have considered that the sur-face on which the adsorption is taking place is not ideal.It contains defects or impurities at random places [28–32], as such the binding strength of particles at theselocations is so negligible that they can not be attachedthere. Except for these places the adsorption is possi-ble if the vacant space is large enough to accommodatea particle. Even in this case, the universal behavior ofthe RSA model is preserved. However, in many realisticsituations, a surface shows spatially correlated proper-ties [33–35] and thus, the existence of spatial correlationsamong the defects is a more natural consideration thanthe randomly distributed defects. Surprisingly, this as-pect has not been considered yet.In this paper, we provide a detailed study on the jam-ming and percolation properties of the RSA model in thepresence of spatially long-range (power-law) correlateddefects. Our main interest is to see whether this spatialcorrelation affects the critical behavior of the transitions.We found that in the regime of non-vanishing spatial cor-relation among the defects, the model does not exhibitthe well-known universal features of the RSA. The criticalexponents associated with the jamming and percolationtransitions are observed to vary systematically with thestrength of the spatial correlation.The remainder of the paper is organized as follows. InSec. II, we describe the model in detail. In Sec. III, wepresent the simulation results on the impact of spatiallylong-range correlated defects on the jamming and perco-lation transitions. Finally, we summarize in Sec. IV. II. MODEL
We consider the RSA model on a two dimensionalsquare lattice of size L × L with periodic boundary con-ditions along both directions. Particles in the form of k -mers, occupying k consecutive lattice sites along hori-zontal or vertical direction, are adsorbed one by one fol-lowing the rules of the standard RSA onto the lattice con-sisting of defects which are spatially correlated. Specif-ically, the defective sites are located in such a mannerthat the correlation function between a pair of defectedsites decays in the form of a power-law of the spacialdistance between them. By imposing this prerequisite,sites of an empty lattice are occupied with probability q and they are kept vacant with probability 1 − q (the de-tailed method for generating such correlated landscape isdescribed later). Initial configuration of these set of oc- cupied sites act as defects and the adsorption of particleson these locations is completely forbidden. Remaining1 − q fraction of sites act as the sites for possible adsorp-tion. By selecting an orientation, either horizontal orvertical with equal probability, one end of the particle isplaced at a randomly selected position and absorbed irre-versibly provided that there exists at least k consecutivevacant sites along the chosen direction from the selectedsite. The surface coverage after adsorbing n particles isthus given by p = nk/L . The adsorption process con-tinues in this way until a jamming state is reached. Thecorresponding surface coverage p = p j is referred as thejamming density.During the process of adsorption, the depositing parti-cles interact among the previously adsorbed particles aswell as with the defects via excluded volume interaction.Consequently, they experience ‘screening effect’ and tryto align parallel to each other and form domains whosetypical sizes are of the order of the size of the particle. Inaddition, due to this interaction, a vacant region that issmaller than the size of the particle can not be occupied.As a result, the jamming density p j can not attain theclose packing density, i.e., p j < − q .As an important step, besides the defect density pa-rameter q , the strength of the correlation among the de-fects is also a tunable parameter in our model and itis characterized by an exponent γ associated with thepower-law decay of the correlation function (see Eq. (1)).For any arbitrary value of 0 < q <
1, the strength of thecorrelation decreases with increasing the value of γ andin the limit of γ → ∞ the scenario of uncorrelated de-fects is obtained. Therefore, by varying the parameters q and γ , the model is capable of capturing the behavior ofa wide range of systems: from a system with correlateddefects, uncorrelated defects to a pure (defect-free) sys- r -4 -3 -2 -1 c (r) FIG. 1. Log-log plot of the height-height correlation function c ( r ) against the spatial distance | r | on a two-dimensional sur-face with L = 2 using γ = 0.4 (black), 0.8 (red), and 1.2(blue) (open circles). By averaging over 10 independent con-figurations, the slope of the fitted straight lines have been es-timated as 0.408(5), 0.804(5), and 1.206(5), respectively (topto bottom). (a) (b)FIG. 2. Typical jamming configurations of dimers on a 96 × q = 0 . γ = 0 .
4. The sites with defects and the absorbed dimers havebeen painted in red and blue colors, respectively. tem. In this paper, we report our simulation results fordimers ( k = 2). Generating long-range correlated defects
To obtain a substrate possesses defects with spatiallong-range correlations, we utilize the idea of viewingthe substrate as a landscape of random heights { h ( x ) } with desired height-height correlation [36], where h ( x )represents the height associated with the lattice site po-sitioned at x . Accordingly, we follow the the schemedescribed in Ref. [37], which is based on the Fourierfiltering method [37–39]. The Wiener-Khintchine theo-rem is the basis of this method, which relates the au-tocorrelation function of a stationary time series to theFourier transform of its power spectrum. The powerspectral density in this case has a power-law form andit is calculated using the two-point correlation function c ( r ) = (1 + | r | ) − γ/ imposing periodic boundary condi-tions in two dimensions. Therefore, c ( r ) decays at largedistance | r | as c ( r ) = h h ( x ) h ( r + x ) i ∼ | r | − γ (1)where, γ denotes the strength of the correlation. Thesteps (i)–(iii) in Ref. [37] are then executed to generatethe correlated random numbers { h ( x ) } .In our simulation, we used Gaussian distributed un-correlated random numbers with zero mean and unitvariance to generate power-law correlated Gaussian dis-tributed random numbers. To verify whether the ob-tained random numbers posses the desired correlationsor not, the configuration averaged value of c ( r ) is plottedwith | r | on a double logarithmic scale in Fig. 1 for threedifferent values of γ . Clearly, the measured slopes of thebest fitted straight lines are consistent with the desiredvalues of γ .Finally, by following the idea of ranked surface [40], thesites are occupied one by one according to the ascending p j / ( - q ) q γ =0.4 γ =0.8 γ =1.2 γ =1.6random FIG. 3. The variation of filling fraction p j / (1 − q ) against thedefect density q for γ = 0.4, 0.8, 1.2, 1.6, and uncorrelateddefects using L = 1024. order of their height values until the density of occupiedsites representing the defects reaches a prefixed value q . III. RESULTSA. Impact on the jamming coverage
In Fig. 2, we have shown typical jamming configura-tions for a given defect density q for both correlated anduncorrelated defects. We first observe that the defectsform clusters via nearest neighbor connections, which be-come more and more compact for stronger correlations(small γ ). An idea of the compactness of the clustersmay quantitatively be realized from the fact that in thesubcritical regime of defect density, i.e., for q < q c , wherethe global connectivity through the clusters of defects isabsent, the average size of the largest cluster of defectsscales with L as: h s defmax ( L ) i ∼ (ln( L )) α (not shown). Itis found that the exponent α > < γ < d , whichmonotonically decreases with increasing the value of γ .This finally approaches α = 1, corresponding to the valuefor uncorrelated defects.Naturally, the particles experience the strongestscreening effect in the case of homogeneously distributeduncorrelated defects during the deposition and it recedesas the strength of the correlation is increased. Thus,we expect to observe densely packed jamming states forstronger correlations. To demonstrate this, the fillingfraction p j / (1 − q ) at the jamming state is plotted against q in Fig. 3 for four different values of γ and for uncorre-lated defects. Clearly, for any given value of 0 < q < γ decreases. L -4 -3 ∆ ( L ) - B FIG. 4. Plot of ∆( L ) − B against L on a log-log scale for γ = 0 .
8. The slope of the best fitted straight lines have beenestimated as 1 /ν j = 0 . , . , . , and 0 . q = 0.1, 0.2, 0.3, and 0.4, respectively (from bottom totop). The corresponding values of B are smaller than B =9 . × − for q = 0 . B. Universality class of the jamming transition
To investigate whether the universality class of thejamming transition is affected by the introduction of spa-tial correlations among the defects, we perform the scal-ing analysis of the width ∆( L ) of the transition zone.Precisely, we calculate the standard deviation of the jam-ming densities ∆( L ) = q h p j i − h p j i for several valuesof L , which generally scales as ∆( L ) ∼ L − /ν j . There-fore, we plot ∆( L ) versus L on a log-log scale. We sim-ulate up to L = 4096 and the averaging are done on (atleast) 5 × , . × , . × , × , and 6 × independent configurations for L = 256, 512, 1024, 2048,and 4096, respectively. It is observed that the curves forsmall γ have certain amount of curvatures and they seemto approach a constant value in the limit L → ∞ . Wethus consider a modified functional form∆( L ) = AL − /ν j + B (2)to fit our data. Indeed, a plot of ∆( L ) − B against L on adouble logarithmic scale exhibits a straight line, as shownin Fig. 4. It is found that B ≈ γ & .
0, but its valueincreases monotonically as γ is decreased. To give anidea, the least-square fit of our data using Eq. (2) yieldsthe values of (1 /ν j , B ) ≈ (0 . , . × − ) , (0 . , . × − ), and (0 . , . × − ) for γ = 0.8, 0.6, and 0.4,respectively, using q = 0 .
2. As opposed to the case ofuncorrelated defects, for which one obtains the universalvalue of ν j = 1 in two dimensions at any arbitrary valueof 0 < q <
1, it is evident from Fig. 4 that the exponent ν j varies systematically with q for a fixed γ .Similar plots are made, but now we keep q fixed andvary γ . To show the effect of the spatial correlation moreexplicitly, we focus on the range of q < q c , where thereexists no giant cluster of defects. It is found that thedeviation of ν j from its universal value ν j = 1 becomes γ / ν j FIG. 5. Variation of the critical exponent 1 /ν j associatedwith the jamming transition as a function of the correlationstrength γ for q = 0 .
05 (black), 0.10 (red), and 0.20 (blue)(from top to bottom). more and more prominent as γ → q . Although, the non-universal behavior is notso obvious from this figure for q = 0 .
05 as the exponentvalue is close to ν j = 1, the curves have apparent cur-vatures on the ∆( L ) vs. L plots for small γ . Precisely,the best fit using Eq. (2) yields a value of B > B ≈ . × − for γ = 0 . q = 0 . q >
0) is sufficient to change the universal behavior of thejamming transition if and only if the defects have strongspatial correlations.What could be the reason behind the origin of this non-universal behavior? One may notice from Fig. 2 that forstrong spatial correlations, the void space becomes frag-mented into several isolated clusters and form islandssurrounded by defects. This happens even for a smallvalue of q (e.g., q = 0 . q = 0 .
1, and the tail of the distribu-tion is observed to shift to the origin with increasing γ (weak correlations). Besides that, the total number ofislands also varies for different configurations. Since theparticle adsorption is being taken place on these islands,one may think that the fluctuation of the jamming densi-ties for a given system size L is a collective contribution percolating dimer percolating defect FIG. 6. Schematic phase diagram in one dimensional q plane.Red and blue dots represent the critical point of percolationthrough the dimers and defects respectively. In the regionbetween the two dots the system percolates neither throughdimers nor through defects. q Π ( q , L ) -2 -1 0 1 2 (q - q cj )L ν p Π ( q , L ) (a)(b) FIG. 7. For γ = 0 .
8, (a) variation of the spanning probabilityΠ( q, L ) of the jamming configuration with defect density q forsystem sizes L = 256 (black), 512 (red) and 1024 (blue) (fromleft to right); (b) finite-size scaling plot of the same data using q cj = 0 . /ν p = 0 . of the fluctuations arising from all those islands. Thus,the variability of the island sizes could be the source ofbreaking the universality class of the jamming transition.Arguably, such a scenario also arises at the percolationpoint of void spaces in the presence of uncorrelated de-fects, where the size distribution of those islands follows ascale-free distribution. However, using extensive numer-ical simulations by setting 1 − q = 0.592746050 (perco-lation threshold of the square lattice), we have obtainedthe universal value of ν j = 1 (not shown). This suggeststhat the departure from the universal behavior for long-range spatially correlated defects is probably not due tothe above mentioned fluctuations and could be related tosome more complex details, such as spatial correlationsbetween the sizes of the islands or the non-trivial inter-actions of particles in the close proximity to the complexinner and outer wall of the compact clusters of defects.Furthermore, we have noticed that the distribution of p j deviates from a Gaussian distribution for strongly cor-related defects for large system sizes. q Π ( q , L ) -3 -2 -1 0 1 2 3 (q - q cj )L ν p Π ( q , L ) (a)(b) FIG. 8. For γ = 1 .
5, (a) variation of the spanning probabilityΠ( q, L ) of the jamming configuration with defect density q forsystem sizes L = 256 (black), 512 (red) and 1024 (blue) (fromleft to right); (b) finite-size scaling plot of the same data using q cj = 0 . /ν p = 0 . C. Percolation transition of the jamming states
We now identify the clusters of absorbed dimers in thejamming state, where a cluster consists of a set of sitesinterconnected through their neighboring sites occupiedby the dimers. For q = 0, the density of occupied sitesis so high ( p j ≈ . q = q c , when a giant clusterof defects first appears in the system, the largest clus-ter of dimers becomes minuscule and it fails to establishsuch global connectivity. Consequently, in between q = 0and q c , one finds a threshold value of q = q cj such thatthe system of dimers exhibits the global connectivity andthus remains in the percolating phase only when q < q cj .In the range of q cj < q < q c , neither the largest clusterof dimers nor defects percolates. The schematic phasediagram in q plane is shown in Fig. 6.The most important question here is, whether the criti-cal behavior of such a percolation transition occurring at q = q cj belongs to the ordinary percolation universal-ity class when the defects have spatial long-range corre-lations. To investigate this, we calculate the spanningprobability Π( q, L ) that there exists a spanning cluster (a) (b) / ν p γ q c j γ FIG. 9. Variation of (a) the critical exponent ν p associatedwith the percolation transition of the jamming states and (b)the percolation threshold q cj as a function of γ . of adsorbed dimers in the system by varying the value of q for three different system sizes L . Then, by performingthe finite-size scaling analysis of Π( q, L ) and estimatingthe scaling exponents we determine the universality classof the percolation transition. Once a jamming configura-tion is reached in our simulation, we check the top to bot-tom connectivity through the neighboring sites occupiedby the dimers using the Burning algorithm [20] imposingperiodic (open) boundary conditions along the horizon-tal (vertical) direction. It may be noted that the dimersadsorbed in the isolated small islands of void space donot help in achieving the global connectivity. Only thelargest island of void space holds a special importance forthis purpose, whose size in its percolating phase scales as h s vac max ( L ) i = ( a + b/ ln( L )) L (not shown). In the con-text of percolation of adsorbed dimers, this may signifyan effective change in the dimensionality of the problem.In general, b > γ increases.In Fig. 7(a), the variation of the spanning probabilityΠ( q, L ) against q has been shown for γ = 0 .
8. By appro-priately scaling the horizontal axis, when the same setsof data are re-plotted against ( q − q cj ) L /ν p we observea nice data collapse [Fig. 7(b)], implying the finite-sizescaling form Π( q, L ) ∼ F [( q − q cj ) L /ν p ] , (3)where ν p is recognized as the correlation length expo-nent of the percolation transition. The analysis yields q cj = 0 . /ν p = 0 . γ = 0 .
8. Wehave also shown similar plots for γ = 1 . q cj = 0 . /ν p = 0 . /ν p = 0 . γ , we see that the critical exponent 1 /ν p increases withincreasing γ and approaches to 1 /ν p = 3 /
4, as shown inFig. 9(a). This dependency is approximately describedby a relation ν p = 2 /γ in the range of γ = 0.6 to 1.0. The p Π ( p , L ) -1 0 1 (p - p c )L ν Π ( p , L ) (a)(b) FIG. 10. For γ = 0 . q = 0 .
1, (a) the spanning probabilityΠ( p, L ) has been plotted against the surface coverage p forsystem sizes L = 256 (black), 512 (red), and 1024 (blue).(b) Finite-size scaling of the same data as in (a) using p c =0 . /ν = 0 . percolation threshold q cj ( γ ) decreases with increasing γ and approaches to 0.3180(5), the value for uncorrelateddefects, as shown in Fig. 9(b). Note that, our resultsfor uncorrelated defects are in good agreement with theprevious numerical data in Ref. [32]. The data used forall these plots are based on averages over (at least) 10 ,5 × , and 7 × samples for L = 256, 512, and 1024,respectively. Therefore, we believe that the above esti-mates are reasonably accurate. D. Percolation transition before jamming
We have seen that for a given value of γ , the defectdensity q = q cj ( γ ) separates between the percolating andnonpercolating jamming states. Specifically, all the jam-ming configurations for q < q cj ( γ ) with density p j ( γ, q )percolate in the limit of asymptotically large system sizes.This suggests that for all values of q < q cj ( γ ) there shouldbe a critical value of p = p c ( γ, q ) such that the systemexhibits global connectivity for p c ( γ, q ) p p j ( γ, q ).In Fig. 10(a), we have plotted the spanning probabil-ity Π( p, L ) against the surface coverage p for three dif-ferent system sizes using γ = 0 . q = 0 .
1. Thesecurves cross each other approximately at a single point p Ω ( p , L ) -1 0 1 (p - p c )L ν Ω ( p , L ) L β / ν (a)(b) FIG. 11. For γ = 0 . q = 0 .
1, (a) variation of the orderparameter Ω( p, L ) against the surface coverage p for systemsizes L = 256 (black), 512 (red), and 1024 (blue). (b) Finite-size scaling of the same data as in (a) using p c = 0 . /ν = 0 . β/ν = 0 . [ p c ( γ, q ) , Π( p c )]. From visual inspection, we estimatethat p c (0 . , .
1) = 0 . p c ) ≈ .
57, whichis quite lower than the value 0.636 454 001 of the cross-ing probability on a cylinderical geometry obtained usingthe Cardy’s formula [41, 42] for defect-free system. It isto be noted that the crossing probability Π( p c ) ≈ . p, L ) against the scaled variable( p − p c ( γ, q )) L /ν exhibits the data collapse for all threesystem sizes [Fig. 10(b)], implying the scaling formΠ( p, L ) ∼ F [( p − p c ( γ, q )) L /ν ] . (4)In percolation problems, average size of the largest clus-ter per site is considered as the order parameter Ω( p, L ) = h s max ( p, L ) i /L , where s max represents the size of thelargest cluster of absorbed dimers. In Fig. 11(a), we haveshown the variation of Ω( p, L ) against p for the samethree system sizes. Again, by appropriately scaling theabscissa and ordinate, and re-plotting the data we ob-serve data collapse of Ω( p, L ), as shown in Fig. 11(b),indicating the scaling formΩ( p, L ) L β/ν ∼ G [( p − p c ( γ, q )) L /ν ] . (5) TABLE I. Our numerical estimates of the percolation thresh-old p c ( γ, q ) for different values of the defect density q andthe correlation strength γ . The numbers in the parenthesesrepresent the error bars in the last digit. γ q p c ( γ, q ) 1 /ν β/ν The finite-size scaling analysis yields p c (0 . , .
1) =0 . /ν = 0 . β/ν = 0 . /ν = 3 / β = 5 / γ, q ) pairs. Interestingly, we found that thecritical exponent 1 /ν depends systematically with q and γ , whereas β/ν always appears to be same as the value ofthe ordinary percolation in two dimensions, i.e., β/ν =5 /
48. The estimated values for different ( γ, q ) pairs arelisted in Table I. For strong correlations ( γ .
6) andhigh q , it is observed that the crossing points of the curvesfor Π( p, L ) vary over a much wider range. In these cases,the two-parameter scaling plot does not exhibit an excel-lent data collapse as seen for γ > .
8. Probably, logarith-mic corrections are responsible for this. Further investi-gations using higher system sizes are thus needed for theprecise understanding of this problem.
IV. CONCLUSIONS
We have investigated the percolation and jammingproperties of the random sequential adsorption of dimerson square lattice in the presence of defects with spatiallong-range correlations. Accordingly, a fraction q of thelattice sites are declared as defects, where the deposi-tion of dimers is completely forbidden. The dimer ad-sorption is taken place randomly at the available vacantspace. The correlation strength among the defective sitesis varied and its impact on the jamming and percolationtransitions have been studied using extensive numericalsimulations.It has been observed that the jamming coverage forany arbitrary value of 0 < q < q is much smaller thanits threshold value q c such that the connected clustersof defects are all minuscule. A continuously tunablevalue of ν j characterizes the jamming transition whichapproaches to its universal value ν j = 1 in two dimen-sions with decreasing the correlation strength.The percolation transition of the absorbed dimerstakes place at a critical density of occupied sites p c onlywhen the defect density is smaller than a critical value q = q cj . The percolation threshold p c has been foundto be dependent on the defect density as well as thestrength of the spatial correlations. For a given defectdensity, p c decreases as the correlation strength is in-creased. Moreover, the finite-size scaling analysis revealsthat the transition does not belong to the ordinary per-colation universality class. The correlation length expo-nent ν changes systematically with the strength of thespatial correlation and approaches to its universal value4 / β/ν associated with the order parameter scaling appears toremain same as the ordinary percolation.Finally, by tuning the defect density q a percolationtransition is observed at q = q cj , which separates the per-colating jamming states from the non-percolating ones.Again, the percolation transition is characterized by anon-universal value of the correlation length exponent,which is found to be dependent on the strength of thespatial correlation among the defects.In the future, apart from the obvious generalization ofthis model by considering different shapes and sizes of theparticles on different lattice geometries or in higher di-mensions, one may find it interesting to study precursor-mediated adsorption in such a correlated disordered en- vironment.We are hopeful that the results presented here wouldprovide a framework for understanding various observa-tions in different experimental conditions more coher-ently, as by tuning the parameters ( q and γ ) of the modela system resembling the real one may be devised. ACKNOWLEDGEMENTS
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