Scaling and width distributions of parity conserving interfaces
We present an alternative finite-size approach to a set of parity conserving interfaces involving attachment, dissociation, and detachment of extended objects in 1+1 dimensions. With the aid of a nonlocal construct introduced by Barma and Dhar in related systems [ Phys. Rev. Lett. 73, 2135 (1994) ], we circumvent the subdiffusive dynamics and examine close-to-equilibrium aspects of these interfaces by assembling states of much smaller, numerically accessible scales. As a result, roughening exponents, height correlations, and width distributions exhibiting universal scaling functions are evaluated for interfaces virtually grown out of dimers and trimers on large-scale substrates. Dynamic exponents are also studied by finite-size scaling of the spectrum gaps of evolution operators.
pacs:81.15.Aa, 02.50.-r, 05.10.Gg, 75.10.Jm
The theory of surface growth processes has by now reached a mature status that allows to describe statistically a wide variety of nonequilibrium phenomena in terms of universality classes of scaling regimes Krug ; Odor ; Henkel . As is known, these latter depend strongly on the conservation laws of the underlying dynamics, bringing about important effects at large times. An interesting example of this, introduced in the context of restricted solid-on-solid dimer growing interfaces Odor2 ; Nijs , is the set of parity conserving (PC) processes Odor ; Henkel . Here, the seemingly microscopic variation of considering the dynamics of extended objects (i. e. involving more than one interface location) rather than that of monomers, has however far reaching implications, giving rise to an anomalous growth of the global roughness or interface width. In one-dimension (1D), where nonequilibrium roughening transitions can also take place Odor2 , this anomaly has been investigated in terms of even-visiting random walks Nijs . In that representation the height levels of the interface are thought of as the visited sites of a 1D Brownian path extended on a given time interval, here playing the role of the substrate length . The constraint to cover each path location an even number of times (or more generally, conserving this number modulo ), introduces long-range temporal correlations Bouchaud which causes the interface to roughen as Nijs . This is in marked contrast to the usual root mean square displacement of normal (diffusive) random walks characterizing the asymptotic width of a variety of interfaces grown out of monomers Krug ; Odor ; Henkel , and typical of both 1D Edwards-Wilkinson (EW) EW and Kardar-Parisi-Zhang (KPZ) KPZ universality classes. Also, roughening anomalies were reported in other growth models with similar global constraints including multiparticle correlations Park , self-flattening, and self-expanding surfaces Park2 .
In this work we examine further aspects of the PC processes referred to above focusing attention on more detailed levels of description, such as height difference correlation functions and width probability distributions. Interestingly and in line with a variety of studies of several growth models Zia ; Racz ; Antal ; Parisi ; Doussal , it will turn out that in approaching the stationary regime there is a single length scale, namely the usual average width, which characterizes these latter distributions in terms of a universal scaling function. Here we follow an alternative description of PC interfaces me using a simple extension of the well known 1D mapping between stochastic dynamics of binary lattice gases and body-centered solid-on-solid (BCSOS) growth processes Krug ; Odor ; Henkel ; Plischke . There, the differences of adjacent pairs of height variables are restricted to , while as is shown in Fig. 1, attachment and detachment of dimers (or -mers in general), are viewed as exchanges of Ising spins on three () consecutive bonds. Although the adsorbed particles do not diffuse explicitly neither in -mer nor monomer form, they are allowed to rearrange throughout the interface by explicit dissociation of -mers. This takes place under desorption attempts which may occur whether the targeted monomers were original adsorbing partners or not. In passing, it is worth mentioning that this is also a typical feature of catalytic surface reactions where the reconstitution of composite objects actually does matter Odor ; Henkel ; Marro .
The simplicity of these rules is deceptive as they entail a number of conservation laws which grows exponentially with the substrate size. At the root of this rather unusual partitioning of the phase space is a useful construction, namely the irreducible string, introduced by Barma and Dhar in closely related systems Barma . We shall exploit this nonlocal construct, defined later on in Sec. II A, using a simple numerical algorithm which enables an approach to the stationary behavior without actually evolving the system. To that aim, we must content ourselves with analyzing just the situation of equal deposition-evaporation rates. The idea, to become clear in a moment, is to concatenate parts of steady configurations of initially flat but small interfaces, such that the final assembly also bears the global constraint of an originally flat but much larger substrate. This approximation circumvents the problem of going through the slow subdiffusive dynamics characteristic of these processes Odor ; Henkel ; Odor2 ; Nijs ; me thus permitting a thorough sampling of width distributions in scaling regimes, and which otherwise would be hard to examine by standard simulations. Nonetheless, to probe simple features of our assembled interfaces under the actual dynamics, we shall also make use of those simulations in large but yet accessible scales. To complement this finite-size approach we also focus attention on dynamic exponents which, as is known Hohenberg , can be read off from the spectrum gaps of evolution operators. Thus, we shall diagonalize these latter exactly in reachable dimensions. In line with what was mentioned before, these exponets will come out to be sub-diffusive and in close agreement with those obtained by the usual dynamic scaling of the interface width Family .
The layout of this work is organized as follows. In Sec. II we recast the master equation of these processes in terms of a quantum spin Hamiltonian through which we not only obtain dynamic exponents but also use to span the entire phase space of small substrates. Following Ref. Barma to identify the conservation laws of this -mer dynamics and utilizing the full configuration lists provided by the spanning, we then put forward our algorithm to sample rapidly the phase space of large interfaces. In Sec. III, at first we test this sampling method so as to retrieve known roughening exponents as well as steady state correlations of smaller systems. Then we carry on with the evaluation of width distributions for which the onset of a universal scaling function is suggested. Surprisingly, this later exhibits a longer tail than that of normal random-walk-like interfaces Zia . Section IV contains a concluding discussion along with brief remarks on open issues and possible extensions of this work.
Ii Dynamics of -mer interfaces
The stochastic dynamics of discrete Markovian systems such as those referred to above, amounts to a generic prescription of transition probability rates between all possible configurations explored in time (here taken as being continuous). Therefore, the evolution of the probabilities to observe the system in one of these latter is controlled by a gain-loss relation known as the master equation Kampen , namely
Conveniently, this relation can also be rewritten in the form of a Schrödinger equation in an imaginary time, that is , thus permitting to derive the probability distribution at subsequent moments from the action of on a given initial condition, i.e. . Here, the Liouville or evolution operator embodying the dynamics is defined through its matrix elements
which, due to conservation of probability, clearly constrain all -columns to add up to zero. Thereby it can be shown Kampen that the steady state corresponds to a unique -eigenmode with eigenvalue , whereas the relaxation time of any observable is upper bounded by , with being the first excitation level of the -spectrum.
In our case, for what follows it is helpful to think of this evolution operator as being applied to a space of -spinors. For that, we interpret the slope configurations of Fig. 1 as being already diagonal in the -component, say, of Pauli matrices assigned to each slope site. By construction it is then clear that up to a constant chosen as a reference level, the heights of the BCSOS interface are obtained as
In particular, note that under periodic boundary conditions (PBC) the dynamics is consistent with a vanishing total magnetization though as we shall see below, many further additional conservation laws also emerge. Introducing now the right and left -mer hopping operators
associated respectively to the detachment and attachment processes described in Fig. 1, and taking into account the algebra of the spin- raising and lowering operators , we can readily write down the operational counterpart of Eq. (2), which here reduces to
Such simplicity is only apparent as the commutation algebra of the hopping operators complicates the analytical treatment (except for monomers and , where reduces to the isotropic Heisenberg ferromagnet). In this latter equation evidently each of the off-diagonal terms provide the appropriate transition elements of the dynamics, whereas the addition of its diagonal parts
accounts for the number of manners in which a given spin configuration may access to other ones either by right or left jumps, i.e. thus complying with conservation of probability. On the other hand combining this with the microscopic reversibility of our model, here expressed simply as (or ) (or ), we then obtain
from which some brief remarks about the steady state distribution now follow. First, note that the monomer case is special in that for PBC (hereafter considered throughout) this latter identity always cancels out as for Eq. (6) simplifies to . Therefore, comparing Eq. (7) with the right hand side of the master equation, we thus see that the monomer steady state, either in equilibrium or not (), is consistent with a constant distribution Derrida . More generally however, and except for the equilibrium situation, this feature does not hold for because the diagonal operators of Eq. (6) are now different, so in general . Since the equiprobability issue is essential for the sampling algorithm that follows, this breakdown will restrict the numerical findings of Sec. III only to the case , yet being non-trivial for as we shall see below.
ii.1 Irreducible strings
Turning to conservation laws and assuming that the lattice is -partite (in 1 just meaning multiple of ), the first invariant set of quantities one can readily identify from the composite spin exchanges of Fig. 1 is that of the sublattice magnetization differences
with . From these possible pairs only of them are independent, so the number of conservation laws would grow at most as . But as mentioned earlier, there is in fact a much subtler set of constants of motion, in turn growing exponentially with the lattice size.
To construct that set, here we briefly survey the ideas of Ref. Barma concerning the dynamics of deposition-evaporation of trimers reconstructing on a line, and which for what follows it is convenient to adapt to larger objects of even length, e.g. . Clearly, those processes are then isomorphic to ours via a simple particle-hole mapping, say on even sublattices. Thus, in analogy to Ref. Barma we now define the irreducible string (IS) of any spin configuration as the sequence obtained by deleting all groups of consecutive anti-parallel spins appearing on chosen locations, and then repeating recursively the procedure on the resulting shorter string until no further of such groups remain. As an illustration, consider for simplicity the following examples of
In the first case this deletion, marked by boxes, is applied to a group of spins chosen either starting from the left or right. In the second instance the procedure is carried out recursively in two steps and no characters are left, while in the third example the string considered is already jammed (same irreducible block of Fig. 1) and can no further evolve. The invariance of the irreducible characters (if any) left by this process is in line with the idea that the successive action of the hopping operators of Eq. (4) on a given spin configuration, just changes the position of those characters by multiples of lattice spacings. The separations between them are mediated by substrings of different lengths (), though all of these are in turn reducible to null strings. Thus, the interface dynamics may be thought of as a random walk of hard-core irreducible characters (they can not cross each other), as depicted schematically in Fig. 2.
The positions of these walkers at a given instant of course depend on the order in which the reduction rule is applied, but the key issue to bear in mind here is that the sequence of irreducible charcaters remains unaltered throughout. In addition and as noted in Ref. Barma , two spin configurations are connected by the dynamics . Thus, the IS uniquely labels all subspaces left invariant by the -mer kinetics, regardless in which order the reducible groups are removed. On the other hand, it is clear that the number of combinations forming these irreducible sequences grows exponentially with the number of characters or string length . More specifically, a straightforward analysis of a recursion relation for this length Barma ; Robin , shows that for large and the number of invariant subspaces increases as fast as , where is the largest root of .
On a more fundamental level, it would be interesting to identify the symmetries at the origin of these conservation laws. For instance, in the much simpler case of the sublattice differences of Eq. (8) the symmetries responsible for them just involve rotation angles around the direction of each sublattice. Recalling that under those rotations transform as , evidently as long as is held, the hopping operators of Eq. (4) will be left invariant, and so will in Eq. (5). Thus, from the infinitesimal generators of these independent rotations one is finally led to the conservation laws of Eq. (8), already obtained on more intuitive grounds. However in the case of the IS and the exponential proliferation of constants of motion it entails, the analysis appears to be much more involved. Due to the highly convoluted form in which the IS is obtained, unlike Eq. (8) it is neither clear how to construct its operational counterpart (possibly non-local), nor to identify the corresponding symmetries in the evolution operator. Despite that formal insufficiency, the invariance of the IS provides an alternative computational tool to approach the equilibrium regime, and which we now implement.
ii.2 Assembling null string states
Notice that whenever is kept finite in the thermodynamic limit, the interface can not roughen at large times me . This is because for the distances between irreducible characters (or random walkers) are distributed as Feller . Thus, mean square height fluctuations along those distances (or reducible substrings, all with ) remain bounded as . So, hereafter we will focus on the null string subspace only. Besides, it is the most natural to consider in the context of growing interfaces, as it stems from initially flat conditions (plain antiferro states).
Although in equilibrium all configurations are equally weighted, yet the expectation value of most observables are not analytically simple to obtain because the ensemble of averaged states must be consistent not just with but also with a vanishing IS. This introduces spatial correlations (absent in the monomer case, where always coincides with ), which develop slowly in the course of growth simulations. At large times however, such process ultimately amounts to producing a uniform distribution of null string states. Here we put forward an approximation to such distribution in large scale substrates based on the construction of small ones. The idea is to assemble pieces of small substrates in such a way that by applying the above deletion rules the whole set is reducible to the null string.
There are several forms to attempt this, but consider for instance a set of spin configurations (sketched as the initial blocks of Fig. 3) with a common length , and drawn randomly from a list of null string blocks previously prepared. This latter, in turn can be constructed from the repeated action of the hopping operators (4) on, say, initial antiferro
states, until exhausting the full space (typically growing exponentially with Barma ). At the first step, half of the drawn blocks are divided in two parts at random locations, while as depicted in Fig. 3, each of the remaining blocks are settled between the split pairs. Thus, one is left with blocks of length all of which are evidently reducible to the null string [ e.g. see Eq. (9) ]. Next, the process is recursed, eventually by splitting further parts of the original blocks, until a single block of length is obtained. The algorithm thus generates an ensemble of substrates which are fully reducible by successive reductions around a central - block, always left unsplit. Though uniform, this assembled distribution (AD) can not be entirely representative of the much larger substrate space, at most it just can be approximative. Nevertheless, as it will be tested out in Sec. III, it does reproduce known features of the scaling regimes in which one is ultimately interested, while enabling to examine there width distributions otherwise difficult to reach. Let us finally comment that had a simple concatenation of blocks been carried out it would have bounded all heights as , whereas on the other extreme, the use of more reduction centers neither would bring about a better approximation to the equilibrium regime.
Iii Numerical results
Before setting out this algorithm in width distributions, we first test it against typical scaling aspects of growing interfaces. In studying these latter one usually considers the mean square fluctuations of the average height , which provides a measure of the global interface width at a given instant, that is
Here, the brackets denote an ensemble average over all possible evolutions of heights, in our case compatible with the null string imposed by flat initial conditions. Based on a wide range of theoretical and numerical studies, it can be argued that should scale as Krug ; Odor ; Henkel ; Family
with a universal scaling function behaving as for , while approaching a constant for . Hence, for the width must grow as note , until saturating as for times comparable or larger than the relaxation time . In the above hypothesis the Hurst or roughening exponent measures the stationary dependence of the interface width on the typical substrate size, while the fundamental scaling between length and time is given by the dynamic exponent .
iii.1 Scaling exponents
When it comes to this latter it is helpful to also consider the spectral gap of the evolution operator constructed in Sec. II, so as to obtain a separate evaluation (independent of ), and which we now briefly touch upon. Assuming as usual the emergence of a finite-size scaling regime in which Hohenberg ,
we analyzed those exponents for and small substrates using a Lanczos diagonalization Lanczos of Eq. (5) within null string subspaces, in turn spanned via the hopping operators (4) as explained in Sec. II B. The results so obtained are displayed in Fig. 4, where it is seen that already modest lengths are able to yield clear finite-size trends which evidence nonuniversal and subdiffusive slopes for dimers and trimers, i.e. 2.61 and 2.89 respectively. In that regard, a convergence estimation of these values can be made by defining the sequence of dynamic exponents or approximants
with . As is shown in the inset of Fig. 4, the relative differences between our highest approximants are about 0.1 % for , and 1.3 % for , which in any case are far apart from the diffusive slope () of the monomer case, only shown for comparison. It would be desirable to improve the convergence of the trimers ’s, but the next approximant () requires to consider spaces of more than null string states which goes beyond our computing facilities. Nevertheless we can compare these exponents with those arising from the standard scaling hypothesis (11) while, more importantly, testing the validity of the type of scheme previously proposed.
To this aim, we compared the evolutions of flat substrates with those resulting from the AD of Sec. II B.. This we do in Fig. 5 where the scaled widths of these two different preparations are displayed for not too large sizes, so as to reach about samples in the final saturation regime (recall that is subdiffusive). The sets exhibit different scaling functions according to the substrate preparations, but in both cases the data collapse was attained upon setting a common roughening exponent (either for and 3), along with common dynamic ones and for dimers and trimers respectively. Interestingly, these two latter values happen to follow closely those of the approximants
referred to above (inset of Fig. 4), while on the other hand, the early time widths arising from the AD already scale around a significant fraction of their asymptotic values, i.e. for dimers and for trimers. This trend still improves when assembling larger - blocks, namely, the scaled widths approach larger fractions of the saturation values observed in Fig. 5 while keeping a common roughening exponent , pretty close to the value obtained above. In Fig. 6 this is corroborated for a variety of substrate sizes () assembled with several - blocks, otherwise unreachable by standard simulations. In that sense note that the algorithm of Sec. II B is not severely limited by the number of recursions (), but rather by the large list of null string states increasing exponentially with . Once these latter are evaluated, the algorithm permits to rapidly average over about samples of rather large lattices.
As we move on towards more detailed levels of description, we now consider the height difference correlation functions for which a similar scaling hypothesis is also expected to hold at distances , that is Krug ; Odor ; Henkel ; Family
As before, the brackets are taken as in Eq. (10) and the scaling function behaves analogously to that of Eq. (11). From this, one infers that a time is required to fully develop the interface roughness across a given distance . Thus, to check out whether our AD can already be associated to late stages of growth, we measured these height correlations using over three distance decades for several - substrates. The results are shown in the upper insets of Fig. 6 where it is observed that in all cases the previous roughening exponent is recovered. For displaying purposes we rescaled these data with the same - amplitudes () used in the main panels, which is no coincidence, as it would be expected on the basis of the identity .
Before continuing we pause to comment on the differences appearing between the exponents of the dimer dynamics and those in the even-visiting random walks (EVRW) analyzed in Ref. Nijs . Our value should not be regarded as a mere numerical deviation from the exponent conjectured in Nijs . At the origin of this departure is the exponential proliferation of irreducible strings appearing in the dimer dynamics which ultimately impose tighther restrictions than those already occurring in EVRW. Although both dynamics share the topological constraint caused by the mod 2 conservation of the number of particles at every height level, note that the BCSOS version of EVRW Nijs mixes up all the many-sector decomposition discussed in Section II A. For instance, besides the dimer dynamics Ref. Park also considers the full restoration of ergodicity by deposition-evaporation of two particles at two randomly chosen columns with equal heights. These need not to be contiguous (as in the dimer dynamics), nor necessarily share the same terrace. That introduces a genuine one-to-one correspondence with the ensemble generated by the EVRW dynamics, so the estimate of found there in such conditions is then in line with the theory of Ref. Nijs . Similarly, that latter work also investigates the effect of adding monomer diffusion within terraces. That produces another estimation which yields . But further to that difference, note that the explicit addition of monomer diffusion partially relaxes the broken ergodicity of the original dimer dynamics. Hence, the above deviations of from should not be ascribed merely to statistical errors but mainly to the change of conservation laws. In fact, when restoring full ergodicity Ref. Park yields a KPZ type exponent which is far apart from the subdiffusive obtained in the original dimer dynamics of both Refs. Nijs and Park as well as in this Section.
Turning to smaller scales and to further probe the AD, we finally compare the exact slope or spin correlations
evaluated in the uniform distributions of our largest available blocks (36 heights for dimers and 48 for trimers), with those estimated in substrates assembled with smaller ’s. This is illustrated in the lower panels of Fig. 6 where it is corroborated that in both cases these functions closely approximate each other. As mentioned earlier on, we thus see that even though the averaged distributions are uniform, the null string constraint enforces non-trivial correlations which otherwise would not appear by the sole restriction of (as in the case of monomers, where all even correlators vanish identically as ). Noting that either Eq. (10) or (13) can also be expressed in terms of , it follows that for these pair correlations are the ultimately responsible for the anomalous roughening of PC interfaces, as opposed to the case where these pairs () have no effect.
iii.2 The width distribution
It is reassuring that both roughening exponents and correlations of Fig. 6, in conjunction with the near-saturated scaling regimes of Fig. 5, all suggest strongly that the construction of Section II B is sampling close-to-equilibrium states. Thus, we carry on and further exploit that construction to compute the width distributions of PC interfaces.
Since the dynamic scaling hypothesis (11) involves in fact an integral over all interface modes, it may well occur that corrections to scaling are needed. In our case, this is particularly noticeable at early stages of evolution where the data collapse in the lower sets of Figs. 5a and 5b is not so evident. In that sense, a variety of theoretical and numerical studies Zia ; Racz ; Antal ; Parisi ; Doussal have suggested an alternative characterization of interfaces in terms of the full probability distribution of its particular random width realizations . So long as their average diverges in the thermodynamic limit, i.e. , the relevance of such distribution relies in that for large substrates it scales as Zia ; Racz ; Antal ; Parisi ; Doussal
where the scaling function is a universal characteristic of the interface fluctuations. Like however, at large times this function can not point out dynamic aspects of universality classes. For example, since in 1D the steady sates grown out of are equiprobable and uncorrelated (recall Sec. II), is the same for both EW () and KPZ () universality classes Zia . But in view of the role of null strings for , should be able to distinguish clearly the growth of PC interfaces from that of monomer ones.
To that aim, note that the basic problem of sampling stationary under very slow relaxation () inevitably arising in standard simulations, is to a large
extent bypassed by the close-to-equilibrium distributions of Sec. II B. In turn, these also enable to sample sufficiently large lattices as otherwise there would be a rather small number of possible values coarse-graining the histogram of . Thus, using substrates in the range of 6.000 to 50.000 heights, in Fig. 7 we display the scaled distributions obtained after dividing the data into intervals (not all shown), and averaging over about samples. This rather extended sampling, facilitated through the algorithm of Sec. II B, was necessary to capture the statistics of the rare events on the tail of the distribution given in the figure inset. As expected, in all cases the data collapse into a single scaling function easily discernible from the exact monomer result Zia . (The collapse is better demonstrated within the scales of the inset where probabilities are much smaller). Furthermore, appears to be the same for dimers and trimers, each stemming from dissimile types of assembled null strings, thus suggesting a universal function for equilibrium PC interfaces. In contrast to the scaling hypothesis (11), here note that there are no parameters to collapse the ’s, and no scaling properties of are used or assumed. The only approximation is the finite size of the systems investigated. In that regrad, other large substrates assembled with different ’s and ’s (not displayed to avoid overcrowding) yielded the same, numerically indistinguishable, functions.
Just as the monomer case, we see that the length scale not only characterizes the macroscopic level of the interface roughness [ Eq. (11) and Fig. 6 ], but also emerges as the natural length of the whole width distribution. Since , one would intuitively presume that height fluctuations in PC interfaces are smaller than those in monomer ones. Thus, on approaching the above length scale one would expect to become more peaked and narrower than the monomer , something which in fact occurs to some extent. However, on the same basis one would also expect the PC distribution to decay faster for large width realizations. Surprisingly, however it turns out to be the other way around. This is illustrated in the inset of Fig. 7 where the semilog plot strongly suggests an exponential decay in the scaled variable rather than its squared, as occurring in the exact solution of Zia . More specifically, the tails of these two distributions behave as
with fitting parameters and . Hence, we conclude that even though the average roughness of finite PC interfaces is significantly smaller () their fluctuations can eventually explore larger widths.
Iv Summary and discussion
To summarize, we have presented an alternative approach to 1D parity conserving interfaces close to their equilibrium regimes (). The notion of irreducible string Barma , which partitions the dynamics into very many disjoint sectors of the configuration space, played an instrumental role in the implementation of the assembling algorithm put forth in Sec. II B (Fig. 3). This latter was shown to provide a fair sampling of the almost saturated state, in turn exhibiting a scaling regime (Fig. 5) controlled by the very same dynamic and roughening exponents obtained through the dynamic scaling hypothesis (11) Family . Without explicitly evolving the system in time, the assembled distribution also reproduced that latter exponent both at the macro scale of the average interface width (main panels of Fig. 6), as well as at the micro level of the height difference correlations of Eq. (13) (upper insets). The value of so obtained is also in excellent agreement with those resulting from simulations in previous studies me and with restricted solid-on-solid versions of these interfaces Odor2 ; Nijs .
As for dynamic exponents, we diagonalized the evolution operator (5) within the accessible null string spaces already stored in assembling the above distributions, so as to analyze the size dependence of its spectral gap (Fig. 4). This provided an estimation of dynamic exponents via a sequence of finite-size approximants (inset of Fig. 4) nearing subdiffusive but nonuniversal values, i.e. and 2.89 for dimers and trimers respectively, both in reasonable agreement with the exponents resulting from the scaling hypothesis (11).
Apart from avoiding the slow subdiffusive dynamics, perhaps the most interesting aspect of our assembling approach is that it also allows for a rich statistical analysis of the full width distribution of large scales. All assembled sizes yielded a single universal scaling function for both dimers and trimers (Fig. 7), in turn quite distinct from that of normal random-walk or monomer interfaces Zia . This contributes to the list of already known scaling functions Zia ; Racz ; Antal ; Parisi ; Doussal which concurrently with roughening exponents may be used to identify static universality classes of growth processes. The only length scale spontaneously emerging in those functions is the average interface width , which in the PC class results conspicuously smaller than in other classes. Curiously, however, height fluctuations in this former turn out to build up in such a way that tails of width distributions decay much slower than those in monomer interfaces (stretched exponential of Eq. (16) and inset of Fig. 7).
Turning back to irreducible strings, note that in equilibrium the only one in which fluctuations diverge even in the thermodynamic limit is just the null string. All other ones containing a finite density of irreducible characters might be considered as non-critical strings. In that regard, Fig. 2 is helpful to understand these latter as random sequences of null substrings of length (distributed as Feller ), through which mean square height fluctuations can not but to remain bounded as . However note that as soon as , stationary probabilities immediately become nonuniform [ recall discussion below Eq. (7) ] and this simple picture no longer holds. Notwithstanding that the nonequilibrium dynamics is still partitioned by the same strings, it remains to figure out whether the current assembling approach, either for null or finite strings, could be extended to incorporate those nonuniform measures. Because of these latter, nonequilibrium width distributions need no longer be related to the scaling function obtained here. Finally, in where there is no analogue of irreducible string, all these issues either in equilibrium or not remain quite open.
It is a pleasure to thank helpful discussions with F. A. Schaposnik. The authors acknowledge support of CONICET and ANPCyT, Argentina, under Grants PIP 1691 and PICT 1426.
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- (26) However for the 1D PC interfaces being considered, the specific form of this scaling hypothesis holds only under detailed balance conditions. Otherwise, the width either exhibits a logarithmic growth or is kept bounded; see Refs. Odor2 ; Nijs ; me .
- (27) See for example, G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, Baltimore, 1996), Chap. 9; Y. Saad, Numerical Methods for Large Eigenvalue Problems, 2nd ed. (SIAM, Philadelphia, 2011), Chap. 6.