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A two-species d-dimensional diffusive model and its mapping onto a growth model

In this work, we consider a diffusive two-species d-dimensional model and study it in great details. Two types of particles, with hard-core, diffuse symmetrically and cross each other. For arbitrary dimensions, we obtain the exact density, the instantaneous, as well as non-instantaneous, two-point correlation functions for various initial conditions. We study the impact of correlations in the initial state on the dynamics. Finally, we map the one-dimensional version of the model under consideration onto a growth model of RSOS type with three states and solve its dynamics.

In this context, simple symmetric (SEP) [8] and asymmetric exclusion processes, in one dimension, (ASEP) [1,2,9] play a particular role because of their relationships with integrable quantum spin systems (Heisenberg chains) and because of their connection with the KPZ equation [10], directed polymers in random media [6], and shock formation (see e.g. [1] and references therein). These models have been extensively studied and the ASEP with open boundary conditions, as a simple driven diffusion model, exhibits a rich dynamical behaviour involving different nonequilibrium phase transitions in the steady states. They can be studied exactly on the basis of the so called Matrix Approach (MA), an algebraic approach based on an ansatz for the probability distribution which is related to the integrability of some quantum spin chains. This approach provides the full solution of the ASEP (and also the SEP) model, including the full phase diagram, density profile and, in principle, any equal time correlation functions. Though, only few explicit results are known [11] about the dynamical correlation functions, much work has been done on the static properties.
The MA has been generalized to solve the stationary states of one-dimensional models with several species [12] and, recently, a first-order phase-transition in some models has been found [13] (see also [14] where different results were obtained, independently, for the same model).
The lack of exact results for the dynamics of multispecies models [15,16] (in particular in dimension d > 1 see also [17] and references therein), has motivated us to study in some details the dynamics of a two-species model, which is related to the models introduced by Arndt et al. in [12,13]. We compute explicitly, in arbitrary dimensions, the density, and the two-point instantaneous/non-instantaneous correlation functions. We then exploit the exact results for the two-point correlation function to study a RSOS-type [6,7,18,19] growth model. The paper is organized as follows: In section II, we present the general stochastic formalism within which we will work. In section III, for the model under consideration, we compute in arbitrary dimensions the density for various initial states and in presence/absence of initial correlations. In section IV, for a translationally invariant version of our model, we evaluate the instantaneous two-point correlation functions in arbitrary dimensions. In particular, in one-spatial dimension, we assume both cases where initial correlations are absent/present. In section V, we introduce and solve a growth model of RSOS-type "with three states". This analysis is carried out for different initial states (correlated and uncorrelated). Finally, in section VI, we calculate for systems with uncorrelated (but random) as well as correlated initial states the non-instantaneous two-point correlation functions.

II. THE FORMALISM AND THE MODEL
Consider an hypercubic lattice of dimension d with N sites (N = L d ), where L represents the linear dimension of the hypercube, and periodic boundary conditions are imposed. Further assume that local bimolecular reactions between species A and B take place. Each site is either empty (denoted by the symbol 0) or occupied at most by one particle of type A (respectively B) denoted in the following by the index 1 (respectively 2). The dynamics is parametrized by the transition rates Γ γδ αβ , where α, β, γ, δ = 0, 1, 2: . For example the rate Γ 12 22 corresponds to the process BB −→ AB, while conservation of probability leads to Γ 11 11 = −(Γ 10 11 + Γ 01 11 + Γ 00 11 + Γ 02 11 + Γ 20 11 + Γ 21 11 + Γ 12 11 + Γ 22 11 ) The state of the system is determined by specifying the probability for the occurence of configuration {n} at time t. It is represented by the ket |P (t) = {n} P ({n}, t)|n , where the sum runs over the 3 N configurations (N = L d ). At site i the local state is denoted by the ket |n i = (1 0 0) T if the site i is empty, |n i = (0 1 0) T if the site i is occupied by a particle of type A (1) and |n i = (0 0 1) T otherwise. It is by now well established that the Master equation governing the dynamics of the systems can be rewritten as an imaginary-time Schrödinger equation: where H denotes the Markov generator, also called stochastic Hamiltonian, and is in general neither hermitian nor normal. Its explicit form is given below. We also introduce the left vacuum χ| which is defined by Probability conservation yields the local equation (stochasticity of where e α denotes the unit vector in the direction α (1 ≤ α ≤ d) and m designates a point of the hyperlattice labelled with help of its d components: m = (m 1 , . . . , m d ) .
In this work, we assume that there are only symmetric nearest-neighbour jump-processes. A particle A (respectively B) can jump, with rate Γ 10 01 = Γ 01 10 > 0 (respectively Γ 20 02 = Γ 02 20 > 0) to an adjacent site (in the d directions) if the latter was previously empty. Such processes are symbolized by the "reaction" A∅ ←→ ∅A (respectively, B∅ ←→ ∅B ). In addition we assume that when two different particles A and B are adjacent, they can cross each other with rate Γ 21 12 = Γ 12 21 > 0. This processes are schematized by the reaction AB ←→ BA. The local Markov generator corresponding to this system and which acts on two adjacent sites m and m + e α reads where the same notations as in reference [21,17] have been used. Probability conservation implies that each column in the above representation sums up to zero. Locally, the left vacuum χ| has the representation χ| = (1 1 1) ⊗ (1 1 1) The action of any operator on the left-vacuum has a simple summation interpretation. This observation will be crucial in the following. Below we shall assume an initial state |P (0) and investigate the expectation value of an operator O (observables such as density etc.) : O (t) ≡ χ|Oe −Ht |P (0) From (3), we can compute the equations of motion of the density and of the two-point correlation functions [17]. For the density, we have: For the derivation of the equation of motion of two-point correlation functions we would proceed similarly, as in [17]. One should however pay attention to distinguish the case of the correlation function of adjacent sites from the general case.
In general, when Γ 01 10 , Γ 10 01 , Γ 02 20 , Γ 20 02 , Γ 21 12 , Γ 12 21 are independent parameters the equations of motions of the multipoint correlation functions constitute an open hierarchy and the dynamics is not soluble. The stationary states of such systems have been studied in [12] by Arndt et al. Recently it has been shown [13] with help of quadratic algebra techniques [13,12] and numerical means that an asymmetric version of this model exhibits a first-order phase transition, in its stationary state, when Γ 01 10 = Γ 10 01 = Γ 02 20 = Γ 20 02 = Γ 21 12 = 1 and Γ 21 12 = q. The steady-state of the density of the same model has also been studied independently by Rajewsky et al. [14] who obtained different results: in [14], the authors, argued that there is no phase transition from the "mixed phase" to a "disordered phase".
Before studying statistical and dynamical properties of this model, let us comment on its solvability. In the single species reaction-diffusion models, the solvability inherent to the closure of the hierarchy was explained in the framework of the duality transformations. In fact, it has been shown that the spectrum of the single-species stochastic Hamiltonian (with the solubility constraints) is identical to the spectrum of an anisotropic spin-1/2 Heisenberg quantum Hamiltonian H XXZ in a magnetic field [20]. As shown in [21], the situation is quite different for the multispecies problem and a general, comprehensive and unified understanding of the formal solubility is still lacking. However, for the model under consideration here, it has been shown [22] that the stochastic Hamiltonian (3), can be mapped, via a similarity transformation, to an exactly integrable quantum spin-1 model introduced by Sutherland [23].

III. EXACT STUDY OF THE DENSITY
In this section, we study the density of the system, in particular, when translation invariance is broken (the initial density is non-uniform) and when correlations in the initial state are present.
It follows from (4) that the density of species j ∈ (A, B) at site m, labelled with its d components (m = (m 1 , . . . , m d )), obeys to the following linear differential-difference equation [17]: We first consider the situation where particles are initially non-uniformally distributed. Namely, we assume that particles of type B are located in the region of space L/2 < x 1 ≤ L, . . . , L/2 < x d ≤ L while particles of type A are initially confined in the region 0 ≤ x 1 ≤ L/2, . . . , 0 ≤ x d ≤ L/2 (we assume that L is even). Within each of the two regions, particles of each type are distributed uniformally with respective densities ρ B (0) and ρ A (0). Solving (6) for this initial condition, we find: where I n (z) denotes the usual modified Bessel function. We have also introduced the Heaviside function Θ(x) = 0 if x < 0 and Θ(x) = 1 if x > 1.
With help of the asymptotic expansion of the Bessel functions and approximating nα e − n 2 α 4Γt ≈ dn α e − n 2 α 4Γt , we obtain from (8) the long-time behavior of the density at site m: where erf (z) denotes the usual error function. Similarly, we have for the density of particles B: We now pass to the case where the distribution of particles for each species j ∈ (A, B) is given by (κ j denotes a real dimensionless constant) an initially correlated distribution: The exact densities then read [17] n j For κ j = 0, in the limit m ∼ L ≫ 1 and Γt where ζ(ν) = k≥1 k −ν , ν > 1 is the Riemann zeta function. It follows from these results that for γ i = 1, the density decays as a power-law of time. However, notice that when initial correlations are strong (i.e. 0 < γ i < 1), the algebraic decay of the density is non-universal (it depends of γ i ). When initial correlations are weak (i.e. γ i > 1), the algebraic decay of the density is universal. Hence the case where γ i = 1 is marginal and there are logarithmic corrections to the universal behaviour.
For initial states decribed by (10), with γ i = 0 and κ j = 0, then we have Therefore the dimensionality has a non-trivial effect: when d < 2, the densities decays as On the other hand, when κ j = 0, the initial density of species j vanishes on the hypercube except at the origin, where its value is ρ j (0). In this case, the limit considered above, yields Notice that because of conservation of the number of particles, in the translationally invariant situation, we simply have

IV. INSTANTANEOUS TWO-POINT CORRELATION FUNCTIONS FOR TRANSLATIONALLY INVARIANT SYSTEMS
In this section we compute exactly the two-point correlation function for translationally invariant systems, in arbitrary dimensions for different initial states.
The equations of motion for the connected correlation functions C ij Solving the latter, we have, using known properties of modified Bessel functions (see appendix,(58)): Restricting the solution to one-spatial dimension, with r ≡ |m − l| ≥ 0, the Laplace transform yields ( i, j ∈ (A, B)) and more generally, Let us now consider one-spatial dimension and assume that the initial correlations are given by We discuss the case |κ l | > 0 while the case κ l = 0 corresponds either to the situation where no particle is present on the lattice initially, or, when all sites of the lattice are occupied by particles of species i (or j). An alternative is that the system would be initially in its steady-state. When a single species is present initially, say species A, we recover the known problem of symmetric diffusion of hard particles A + ∅ ↔ ∅ + A. When the lattice is full (or empty) initially, no dynamics takes place. A single-species one-dimensional process A + A ↔ ∅ + ∅ with a correlated initial state as in (20) has been studied in [18]. Again we can infer the asymptotic behavior of the two-point connected correlation functions in the limit Γt ≫ 1 with v ≡ L 2 8Γt < ∞ [17]. It is useful for the sequel to introduce the definitions of the auxiliary functions [17]: We distinguish two regimes i) For r ≪ L, with r 2 α /8Γt ≪ 1 and σ ≡ r L , ii) For r ≫ 1, r ≡ σL ∼ L, we have As for the density, it follows from these results that for ν l = 1, the (connected-)correlation functions decay as a power-law of time. When initial correlations are strong (i.e. 0 < ν l < 1) the power-law decay of the correlation functions is non-universal . In contrast, when initial correlations are weak (i.e. ν l > 1), the algebraic decay of the correlation functions is universal. The case where ν l = 1 is marginal and logarithmic corrections to the universal behaviour arise.

V. THE MAPPING ONTO A RSOS GROWTH MODEL WITH "THREE STATES"
In this section, we introduce a growth model of RSOS type (Random Solid on Solid ) with "three states", by exploiting a mapping of the one dimensional model studied in the previous sections.
Let us briefly recall that the RSOS growth models are e.g. useful to describe the spatial fluctuations of the (onedimensional, of length L) interface location in the magnetization profile between coexistent phases in two-dimensional models of ferromagnets, such as in the zero field planar Ising model [2]. In such models, at zero temperature, every path minimizing the energy of the systen is a sequence of L binary numbers n j = 0, 1 with j = 1, . . . , L. The stochastic variable n j has the value n j = 0 if the j th segment of the interface steps upwards (in an angle of π/4). The value n j = 1 corresponds to the case where the segments steps downwards with an angle of π/4. The quantities n j = 0, 1 can be interpreted as occupation numbers relating the interface height h j according to h j − h j−1 = 1 − 2n j [2]. In this case the displacement δh r (t), at time t, of the segment of the interface from the sites j 1 to j 2 > j 1 , with r ≡ j 2 − j 1 > 0 is given by δh r (t) = r k=1 (1 − 2n k (t)) and thus |δh r+1 − δh r | = 1.
Here we consider an extension of the above model. We consider that the configurations minimizing the energy (at zero temperature) are of the form where the discrete stochastic variable x A n A j + x B n B j can take three values. The case x A n A j + x B n B j = 0 again corresponds to the situation where the j th segment steps upwards with an angle of π/4. The case x A n A j + x B n B j = x A (respectively, x A n A j + x B n B j = x B ) describe the situation where the j th segment forms an angle Arctan(1 − x A ) (respectively, Arctan(1 − x B )) with the horizontal. When x B = 0 (respectively x A = 0) and x A = 2 (respectively x B = 2), we recover the above-mentioned two-state RSOS growth model. In this sense the growth model which we study hereafter is a "three-states" extension of the usual RSOS model [2,7,18,19]. In addition the mapping with the diffusive model is clear: the presence of particle of species A (resp. B) at site j translates in the language of the growth model with the fact that the related segment of the interface forms an angle Arctan(1 − x A ) (respectively, Arctan(1 − x B )) with the horizontal. In this picture, the jumping of the diffusive particles correponds to the fluctuation of the orientation of the related segments of the interface.
We consider a translationally invariant system; the displacement of the (one-dimensional) interface, at time t, from the sites k 1 to Therefore, in the model considered here, we have max|δh instead of the usual constraint |δh r+1 (t) − δh r (t)| = 1 of the conventional RSOS models. The mean-displacement of the interface reads δh thus , if one wants to impose a zero mean-displacement of the interface , we have to require that In this section we are interested in the computation of the fluctuations of δh j (t): where w(r, t) is the physical width of the interface .
Using the fact that ∂ ∂t r l=1 n i m n j m+l (t) = 2Γ n i m n j m+r+1 (t) − n i m n j m+r (t) and r s=1 , we obtain the following equation of motion for the width: For r > 1, we thus have: where With help of the formula [24]: , and with the explicit expression of the correlation functions (18) and (19), we obtain the following exact expression for w(r, t): We will now specifically focus on two kinds of initial states: (i) We assume first that the system is initially characterized by an alternating periodic array of particles of type A and B. We consider thus an initial state |P (0) = 1 2 (|ABAB . . . + |BABA . . . ), with x A + x B = 2 (in this case δh r (t) = 0). This initial flat interface leads to ρ A = ρ B = 1/2, and for the connected initial correlation functions, we have: C AA r (0) = (−1) r 4 = C BB r (0) = −C AB r (0). It follows from (31) that the initial fluctuations read in this case: w 2 (r, 0) = r 4r + 3 4 (x A − x B ) 2 − 1 , for r > 1 and r even. Therefore, we also have w 2 (1, t) = 3.
For this initial configuration, the expression (18) simplifies and we have C AA The general expression (32) of the fluctations reads: . From this expression, using the asymptotic behavior of the Bessel functions, it is possible to obtain the long-time behavior (for Γt ≫ 1 r ≫ 1) of the fluctuations Thus for the initial condition considered here, when x A = x B , it follows from (33) that the fluctuations growth as w(r, t) ≈ (Γt) 1/4 . On the other hand, it is known that for the flat interface (or "sawtooth" initial state), which dynamics is coded in a two-state model δh r = n m=0 (1 − 2n A m ), the fluctuations growth as ∼ (Γt) 1/4 [7,18,2]. In the situation considered here, the fluctuations still grow as w(r, t) ∼ (Γt) 1/4 and thus the details of the model and its "three-state" character only appears through the amplitude (x A − x B ) 2 . When x A = x B = 1, the initial configuration corresponds to a straight line and, according to (33), there are no fluctuations. Let us also note that when, e.g., x A = 2 and x B = 0, the B particles play the role of the vacancies in the two-state model and the model (27) is exactly mapped onto the well studied two-state model [7,18,2].
(ii) We now investigate the fluctuations of δh r (t) in the presence of initial correlations. We assume that particles of type A and B are distributed according to [18] C l r (0) = κ l r −ν l , ν l ≥ 0, l ∈ (AA, BB, AB) which corresponds, via the mapping (27), in the language of the growth model, to an interface with initial fluctuations given by (31), where n i m n j m+r ′ (0) = C ij r ′ (0) + ρ i ρ j , (i, j) ∈ (A, B) Let us define the following quantity: With help of (29) and (20), (23), we can compute th asymptotic expression of the fluctuations for Γt ≫ 1 and r ≫ 1, with v ≡ L 2 8Γt and σ ≡ r/L = O(1) (we assume that x A x B = 0), which reads where the quantities F 1 and F 2 have been defined in (21), (22) and ξ(ν) is the usual Riemann zeta function. We see that in the presence of initial correlations (34), the fluctuations are dominated by the smallest initial correlation exponent ν. Therefore, if ν = ν AA (resp. ν = ν BB ), the dominant contribution (36) to the fluctuations are the same as for a correlated two-state model where x A = 2 and x B = 0 (resp. x A = 0 and x B = 2) where the B (resp. the A) particles play the role of vacancies [18].

VI. NON-INSTANTANEOUS TWO-POINT CORRELATION FUNCTIONS
In this section, we compute exactly the non-instantaneous two-point correlation functions for various initial states. Similar quantities have already been computed, for some specific single-species models (see e.g. [18,[25][26][27]).

Let us first consider an uncorrelated initial distribution |P
We then have : We are interested in the asymptotic behavior (Γt ≫ 1 and u = L 2 /4Γt < ∞) of the above functions in the two regimes: It is worth noting that the autocorrelation functions are obtained in the second regimes (ii). We then have In these regimes, we have a power-law decay of correlation functions (i, j) ∈ (A, B)), namely, Let us now pass to the case where the initial state is correlated according to (45) and Notice that in one dimension the initial state (45,46) is translationally invariant and reads: This translational invariance which is broken (see (45)) in higher (d ≥ 2) dimensions leads to two regimes: i) We begin with the one-dimensional case (d = 1), here r = r α ≡ |m − l|. Because of the initial translational invariant state, we expect that the non-instantaneous correlation functions only depends on r = m − l, and we obtain n A

VII. SUMMARY AND CONCLUSION
In this work we studied, by analytical methods, the dynamics of a symmetric two-species reaction-diffusion model in arbitrary dimensions. We mapped this model onto an one-dimensional RSOS-type growth model and obtained explicit results for the latter. In particular, we were able to compute the density profile for three various initial conditions (uniform and non-uniform) in arbitrary dimensions.
Furthermore, we evaluated, for a translationally invariant system, the instantaneous two-point correlation functions in arbitrary dimensions. In one-spatial dimension, we considered the case where initial correlations were present. We observed that when the initial correlations are strong enough, they affect the asymptotic dynamics. We also noticed a crossover in the dynamics between the case of strong and weak initial correlations.
We mapped the one-dimensional version of the reaction-diffusion model on a "three-states" RSOS-type growth model. Using the exact, instantaneous correlation functions, we computed the exact expression of fluctuations of the interface for the latter model. We specifically considered the case of a "flat interface" where the fluctuations growth as (Γt) 1/4 , as in the corresponding "two-state" growth model: the three-state nature of the model considered only appears in the amplitude. We also considered the case where the initial configuration is translationally-invariant, random and correlated. We saw that the initial correlations are "strong", they affect the long-time behavior of the fluctuations of the displacement interface. Conversely, "weak" initial correlations, do not affect the dominant term and the fluctuations still grow as (Γt) 1/4 . This is analog to what happens in the two-state RSOS systems where initial correlations affect the long-time behaviour of the width [18].
Finally we computed in arbitrary dimensions the exact non-instantaneous two-point correlation functions for initially uncorrelated states as well as for cases where correlations were present. Here we again observed the effect of strong initial correlations on the dynamics and a crossover between regimes with strong and weak initial correlations takes place.
We conclude this work by addressing an interesting question based on the similarity of the stochastic Hamiltonian under consideration with the integrable Sutherland's quantum spin system [22,23]. It is known that for one-dimensional SEP-model, the relation with the Heisenberg chain has been fruitful to obtain a (dynamical) matrix formulation of the probability distribution, which allowed to solve the (dynamical) density profile for a SEP-model with open boundary conditions (particles were injected and ejected from both ends of the chain) [8]. The relation of the present model to Sutherland's one suggests that a dynamical matrix approach would be possible (in one-spatial dimension) to treat the injection and ejection of particles at the boundary (open boundary conditions).

VIII. ACKNOWLEDGMENTS
The support of Swiss National Fonds is gratefully acknowledged.

IX. APPENDIX: SOLUTION OF THE EQUATIONS OF MOTION OF THE INSTANTANEOUS TWO-POINT CORRELATION FUNCTIONS
Solving the equations of motions (16) for the instantaneous correlation functions in arbitrary dimensions, for the model under consideration in section V I, we obtain the following expression: Using the properties of the derivatives of Bessel functions and then integrating by parts, we obtain the more compact form (17)