Jamming and percolation of dimers in restricted-valence random sequential adsorption

Restricted-valence random sequential adsorption~(RSA) is studied in its pure and disordered versions, on the square and triangular lattices. For the simplest case~(pure on the square lattice) we prove the absence of percolation for maximum valence $V_{\rm max}=2$. In other cases, Monte Carlo simulations are used to investigate the percolation threshold, universality class, and jamming limit. Our results reveal a continuous transition for the majority of the cases studied. The percolation threshold is computed through finite-size scaling analysis of seven properties; its value increases with the average valency. Scaling plots and data-collapse analyses show that the transition belongs to the standard percolation universality class even in disordered cases


I. INTRODUCTION
Percolation [1] is characterized by the formation of a spanning cluster in a system composed of elements (sites and/or bonds) each present independently with probability p. The probability of a spanning or percolating cluster is only nonzero for p > p c , the percolation threshold, marking a continuous phase transition with associated critical exponents [2][3][4].
Random sequential adsorption (RSA) [2,14] is a stochastic process consisting in irreversible deposition of immobile objects onto an initially empty substrate, such that each object excludes a certain area from further occupation. A realization of RSA stops when no further deposition events are possible, at which point the system is said to be jammed.
Introducing a deposition attempt rate (per unit area, or per site, on a lattice) of unity, a time can be associated with each deposition event in a given realization. Letting the coverage ρ denote the fraction of the substrate occupied by deposited objects, we define ρ (t) as the mean coverage (over all realizations) at time t. Of particular interest is the jamming or saturation coverage ρ ∞ ≡ lim t→∞ ρ (t). RSA of objects of diverse formats (discs [15], linear k-mers [16,17], etc.) has been used to model a wide range of physical processes such as ion implantation in semiconductors [18], protein adsorption [2], as well as the original car-parking problem [19].
Consider RSA on a two-dimensional lattice, of objects occupying two or more sites. As the coverage increases, it may be possible for the set of deposited objects to "percolate," i.e., to form a spanning cluster, in which case the mean coverage at percolation is a quantity of interest. (While RSA of monomers -objects occupying a single site -is trivial, percolation of monomers is a classic problem and p c has been determined, exactly or numerically, for a wide variety of lattices [20][21][22][23][24][25].) RSA of extended objects, such as rods or linear lattice k-mers, is a problem of current interest. Cherkasova et al. [26] showed that the percolation threshold for dimers (k-mers with k = 2) is smaller than for monomers. As expected, the universality class does not depend on the length k. In addition, these authors observed that when the dimers can only align along one direction, the percolation threshold increases.
Cornette et al. [27,28] studied the influence of k in the percolation threshold via Monte Carlo simulation (MC) and Bethe lattice analysis. They confirmed that p c decreases mono-tonically with k for 1 ≤ k ≤ 16, and that the universality class of the transition is standard percolation, independent of k. Leroyer and Pommiers [29] demonstrated that the percolation threshold p c decreases with increasing segment length, reaches a minimum value and then increases for k ≥ 15. Tarasevich et al. [30,31] conjectured, based on simulation, that for k-mers of sufficient length (k 1.2 × 10 4 ) percolation does not occur. More recently Kondrat et al. [32] developed a rigorous proof refuting this conjecture. They showed that for nonoverlaping k-mers, the jammed configuration includes a percolating cluster.
Although deposition of particles or k-mers on a regular substrate has attracted much attention, this model needs to be extended to describe more realistic situations. In many systems, the substrate includes impurities and/or defects that affect the deposition process.
A simple realization of substrate disorder involves excluding randomly a fraction of sites from the deposition process, resulting in a diluted system. For example, Cornette et al. [33] studied deposition of polyatomic structures on diluted lattices, observing that the percolation threshold increases with dilution, and that dilutions greater than a certain value, there is no percolation [31,33].
In standard dimer RSA, a given site can be occupied by at most one monomer, and only one dimer can be incident upon an occupied site. A way of softening this restriction, while still prohibiting arbitrarily high densities, is to allow up to V max dimers to be incident on a given site. In the resulting restricted valence RSA process, each lattice edge can be occupied by at most one dimer. Here, V max ranges from unity (the usual dimer RSA problem) up to z, the lattice coordination number. (In the latter case, the RSA process is trivial since all edges are eventually occupied.) Restricted-valence RSA is analogous to a self-avoiding random walk (SAW) in which the walk is allowed to visit the same site up to K times.
Krawczyk et al. [34] showed that existence and nature of phase transitions in these models depends on the details and dimensionality. For example, in case of forbidden reversal on the cubic lattice, discontinuous and continuous transitions are observed, with the continuous transition of the same type as in interacting self-avoiding walk collapse (ISAW) [35]. On the other hand, for allowed reversal on the square lattice, these authors do not find any indication of a phase transition. Oliveira et al. [36] studied SAWs with K = 2, using Husimi and Bethe lattice solutions. They found a rich phase diagram with regular polymerized, nonpolymerized, and pair polymerized stable phases, a tricritical point and a critical endpoint. The transition between polymerized and nonpolymerized phases can be continuous or discontinuous depending on the region of the phase diagram.
We consider restricted-valence RSA (and the associated percolation problem) on the square and triangle lattices, in both its pure form (all sites having the same maximum valence) and with disorder such that the the maximum valences at each site are independent, identically distributed random variables. Our objective is to understand how percolation thresholds and jamming limits depend on the valence restrictions. We verify that percolation transitions, when they exist, belong to the standard percolation universality class. In addition, we develop a proof for the absence of percolation for maximum valence two.
The remainder of this paper is organized as follows. In Section II we detail the restrictedvalence RSA model. Simulation methods are presented in Sec. III followed by results in Sec. IV. Finally, in Sec. V, we present our conclusions. The proof is given in the Appendix.

II. MODEL
We study RSA of dimers on a regular lattice under the restriction that the number of dimers that can attach to a vertex (its valence) cannot exceed V max . In case V max = q, with q the coordination number of the lattice, there is no restriction and all edges of the lattice are eventually covered by a dimer. The cases V max = 1, 2 and 3 on the square lattice are shown in Figs. 1 (a), (b) and (c) respectively. For V max = 1, we have the usual irreversible dimer or domino tiling problem [37], in which only isolated edges may be occupied (see Fig. 1(a)).
For V max = 2, a vertex may have a maximum of two incident edges, giving rise to open or closed nonbranching paths, as shown in Fig. 1(b). Fig. 1(c) shows a typical configuration for V max = 3. In Fig. 1   half the edges have been visited in the RSA process. The process terminates when the system is jammed, which happens for some (sample-dependent) p final ≤ 1.

III. SIMULATION
We use a variation of the union-find algorithm of Newman and Ziff [38,39] (NZ) to generate the RSA configurations efficiently and to estimate the percolation point. In this algorithm, we first create a list of pairs of neighboring vertices associated with each edge on the lattice. For each realization of the RSA process, we generate a random ordering of this list. In the usual NZ algorithm, we go down this list one pair at a time and add the bonds to the system, using the "find" routine to find the roots of the two clusters at the ends of the new bond, and then the "union" step to join two clusters if their roots are currently marked as distinct. Ordering the bonds beforehand is very useful for the RSA problem since we can just go down this list, thus considering each bond just once. For the restricted-valence models, we modify this program to keep track of the valence of each site, and only occupy a bond if the valences of its two sites are < V max .

IV. RESULTS
In this section we report results the for the percolation threshold, critical exponents and jamming coverage for restricted-valence RSA on the square and triangle lattices. The properties of interest are exhibited as functions of the control parameter p, i.e., the fraction of bonds visited. In percolation, the order parameter is usually defined as [40] Ω (p, where s max is the mean fraction of sites in the largest cluster, d is the dimensionality, and the angular brackets denote an average over realizations. The percolation threshold of the infinite lattice p ∞ c is estimated via finite-size scaling (FSS) analysis [1] and is expected to follow, where p c (L) is the pseudocritical value for lattice size L and ν is the critical exponent governing the correlation length. The pseudocritical value is commonly determined through the position of the maximum of some "diverging" quantity, or of a crossing point for different system sizes. In this work we analyze a set of seven quantities; five are moment ratios of the form, where the moments M k are defined as M k (p, L) ≡ s k max and k = 1, . . . , 4, We also study the second cumulant K 2 (p, L), This property, also called the susceptibility χ (p, L), provides information about the fluctuations in the size of the largest cluster. Finally, we analyze M 2 (p, L), defined as the difference between the second moment of s M 2 (p, L) and the mean fraction of sites in the largest cluster: where In Eqs. (5) and (6), s denotes cluster size and n s the average number of clusters of size s.
The FSS theory of percolating systems [41] states that at the critical point p = p ∞ c , Ω (p, L), M 2 (p, L) and K 2 (p, L) obey the relations, and where β, γ and ν are critical exponents and ε ≡ p − p c represents the distance to the critical point. The scaling functions U(x), M(x) and K(x) exhibit universal behavior.  the global maximum. (The order of the polynomial is chosen as the lowest order that yields residuals without systematic behavior. In most cases fourth-order polynomials are used).
The uncertainty in the position of the maximum was estimated through the root mean square deviation (RMSD); in the worst case, the RMSD ∼ 10 −4 (see Table I).
In the limit L → ∞ all the estimates for the percolation threshold converge to very similar values (see Fig. 4 and Table I). The final estimate for the percolation threshold,  with the theoretical prediction of the ratio γ/ν for SPUC. For K 2 (see Fig. 5(c)), the data points and the log-log plots (see inset) provide a slope of 1.782 (8), again in agreement with the exact result γ/ν = 1.7916 . . . [1]. The ratios β/ν and γ/ν satisfy the hyperscaling relation dν = γ + 2β within uncertainty.

Valence disorder
We now analyze restricted-valence RSA with valence disorder, in which the valences associated with each site are taken as independent, identically distributed random variables. There is no percolation for cases (1,3) and (1,4). While disorder alters the percolation thresholds and jamming coverages, it does not appear to affect the critical exponents. According to the Harris criterion [42,43], disorder is relevant if dν ≤ 2. Since the transition in the pure system belongs to the SPUC with dν = 2(4/3) > 2, disorder is indeed expected to be irrelevant. The quantities K 2 (p, L), M 2 (p, L) and Q 11 11 in the presence of disorder are shown in Figs. 6(a)  Note as well that the crossing value of Q 11 11 is the same as for the pure case. As expected, the percolation threshold moves to lower values of p as the mean valence increases. For example, for (3, 4) (blue lines in Fig. 6(a), (b) and (c)), half the sites have maximum valence three and half maximum valence four. Both valences percolate and the percolation threshold occurs at a value between those for the pure cases.
Our estimates for the percolation threshold are obtained via FSS as shown in Fig. 6(d), providing p ∞ c = 0.7283(5), 0.6099(1), 0.5120(2) for the cases (2, 3), (2,4) and (3,4) respectively. The analyses follow the same lines as the pure cases (3,3) and (4,4). It is worth noting that although (1,4) and (2,3) possess the same mean valence, the latter percolates while the former does not. Results for the coverage density ρ as a function of p and the jamming coverage are shown in Fig. 8. Figure 8(a) exhibits the density of occupied bonds as a function of p. The red dashed line represents the limiting case (4,4) in which all bonds are occupied. While the curves are generally similar, those for (1, 4) and (2, 2) cross near p = 0.60, suggesting that the rate at which bonds are occupied changes in a nontrivial manner as p varies. In Fig. 8(d) we plot the jamming density versusV , the arithmetic average of the valencies in the range {V i , . . . , V j }; p j increases approximately linearly withV .

Percolation thresholds
We determine the bond occupation fractions at the pseudocritical percolation points, ρ c (L), which, when extrapolated to infinite system size, furnish estimates for the bond percolation densities, ρ c , for the different cases of pure and mixed valences (see Table II).
Pure valence-4 is simply bond percolation on the square lattice and our result is consistent with the exact value, ρ c = 1/2 [4]. At the other extreme, (2,3), ρ c is clearly larger than one-half. Recalling that the pure valence-2 system does not percolate, ρ c > 1/2 can be understood qualitatively by noting that many occupied bonds falling in regions rich in valence-2 sites cannot contribute to percolation. A similar observation applies to the (2,4) mixture. For (3,3) the deviation from ρ c = 1/2 is barely significant, while for (3,4) our result is consistent with a percolation density of 1/2. It is nonetheless surprising that the percolation density for (3,3) is smaller than 1/2. While we defer a detailed study of this case to future work, we note that in all cases except (4,4), the occupation of bonds incident upon a given site are not independent events, so that deviations from standard independent percolation are possible in principle.
We summarize our results for the percolation threshold, jamming coverage and the density of occupied sites at jamming for the square lattice in Table II.

B. Triangle lattice
We adopt the same approach as employed for the square lattice; the scaling behaviors are quite similar, hence we only report numerical values. For (1, 1), (2,2), (1,2) and (1, 3), valence restrictions prohibit percolation. All other cases exhibit a percolation transition characterized by size-dependent peaks in M 2 , K 2 , and the Q js ir . Table III exhibits simulation values for case (4,4).  In the limiting case (6,6), the percolation threshold is known exactly: p c = 2 sin(π/18) = 0.347296355... [20]. Our estimate of p ∞ c (6, 6) = 0.3472(5) is in agreement with this result. Estimates for the other cases are shown in Table IV. Analysis of critical exponents again yields values consistent with SPUC. For example, scaling plots of M 2 and Ω using SPUC exponents exhibit crossings of curves at the percolation threshold.
The jamming probability and coverage density are determined via the same techniques used for the square lattice. The curves for the density of occupied bonds for (2,5) and (2,6) cross the curve for (3,3), similar to the cases (1, 4) and (2, 2) on the square lattice.
A compendium of results for the triangle lattice is reported in Table IV. Different from the square lattice, here all percolation thresholds are greater than or equal to the exact value for the unrestricted triangle lattice. (1)

V. CONCLUSIONS
We investigate restricted-valence random sequential adsorption in its pure and disordered versions, on the square and triangle lattices. We show that there is no percolation for V max = 2 on the square lattice. In other cases, Monte Carlo simulations coupled with the Newman-Ziff algorithm are used, revealing a continuous transition for cases (2,3), (2,4), (3,3), (3,4), and evidently, (4,4). Finite-size scaling analysis of M 2 , K 2 and Q js ir is employed to estimate the percolation threshold. Scaling analyses show that the critical exponents are in good agreement with the standard percolation universality class, as one might expect given that the correlations are local. In the disordered cases, the universality class is preserved although the percolation threshold naturally depends on the average valence.
The coverage densities and jamming coverages are estimated for all cases. On the square lattice, our result for the jamming of dimer RSA, θ = 0.906814 (5), is in agreement with previous estimates of Nord and Evans [44] and of de Oliveira and coworkers [45], and surpasses their precision. The jamming fraction was estimated for all cases, with uncertainties on the order of 10 −6 . Our estimates for percolation thresholds in restricted-valence cases (i.e., some fraction of sites having V max < 4) are ≥ 1/2, as expected, except for (3,3), where we find p c = 0.496 (2). For the triangle lattice, our estimate for p c is again in agreement with the known exact value. The largest estimate for p c is for (1,4), with p ∞ c = 0.8582 (5). Our study also provides an estimate for the jamming coverage of dimer RSA on the triangle lattice: θ = 0.914028 (2). This is consistent with, and considerable more precise than, the previous estimate of 0.9142(12) obtained by Perino et al. [46]. Our results for the triangle lattice are again consistent with the standard percolation universality class.  Consider a final configuration of dimer deposition on Z 2 with V max = 2. Let C be the set of sites connected to the origin, O. C is the union of sites that are 1, 2, ..., n steps from O, along the path of occupied edges linking the sites to O. Let X j ∈ {0, 1, 2} be the number of sites exactly j steps from O in C. (In case C be a closed loop, we take j as the minimum number of steps to O.) We note that X j is a nonincreasing sequence, with X j = 0 for j > n. Let P (n) be the probability, over the space of all final configurations, that the maximum number of steps from O in the cluster containing the origin is n. We aim to show that ∃ c < 1 such that P (n) < c n , so that E n < ∞, i.e., there is no percolation. Let A j be the event X j = 2. We show that P (A j+1 |A j ) < 1. Given A j , let x j be one of the sites that are j steps from O, let z 1 , z 2 , and z 3 be the sites neighboring x j distinct from x j−1 , and let y i be the edge linking sites x j and z i . Let B j be the event that edges y 1 , y 2 , and y 3 are all unoccupied in the final configuration. Occurrence of B j implies that X j+1 < 2.
We now argue that P (B j ) > 0 for any j > 0. A sufficient condition for occurrence of B j is that sites z 1 , z 2 , and z 3 are all of valence two, via edges y m / ∈ {y 1 , y 2 , y 3 }. An example of such a configuration is shown in Fig. 9. Recall that each edge y is assigned a real time t y ∈ (0, 1]. Initially all edges are empty, and edge y becomes occupied at time t y if permitted by the maximum-valence rules; otherwise it remains empty for all time. Consider a site s and let u 1 , ..., u 4 be the edges incident on s, with t 1 , ..., t 4 their associated times. Call e(s) the pair of edges u i having the smallest and second smallest times. For an arbitrary edge y, let s 1 and s 2 be its terminal sites. A sufficient (but not necessary) condition for y to be occupied in the final configuration is that y belong to e(s 1 ) and to e(s 2 ). Using the fact that the t j are i.i.d. uniformly distributed on (0,1], one readily verifies that Prob[y ∈ e(s 1 ) ∩ e(s 2 )] = 13/35. Now consider the local configuration shown in Fig. 10. A sufficient condition for the edges indicated by solid lines to be occupied is that they belong to the sets e associated with the three sites upon which these edges are incident; these conditions naturally imply certain restrictions on the t i . Integrating over the t i , considering the diverse orderings allowed by the above-mentioned constraints, we find the probability of the two-edge configuration shown in Fig. 10 to be at least 87/900 ≡ b 1/3 . It follows that P (B j ) > b > 0. By the same argument, if X j = 1, there is a nonzero probability that X j+1 = 0. Thus there is a finite constant C such that Prob[X n > 0] < C(1 − b) n implying that the percolation probability is zero. Although the argument is for the square lattice, it can be adapted via straightforward modification to arbitrary lattices of finite degree.