Growth fluctuations in a class of deposition models.

*(English)*Zbl 1029.60075The author considers a large lass of deposition models, which are stochastic growth models whose growth mechanism is locally (i.e., microscopically) defined. The author obtains a law of large numbers for the global behavior and – more interestingly – a central limit theorem with explicit identification of the asymptotic variance. The model considered in the paper is more general than models for which these questions have already been studied in the literature, like the totally asymmetric simple exclusion process, the zero-range process or the Bricklayers’ model.

The model is as follows. Let \(I\) be a (bounded or unbounded) interval of integers, and consider the following random process on the state space \(\Omega=I^{ \mathbb Z}\). Assign to each edge \((i,i+1)\) the height, \(h_i\), of a column built of bricks which lie on the edge. The configuration \(\omega=(h_{i-1}-h_i)_{i\in\mathbb Z}\) is the collection of negative gradients of the heights of the ‘wall\'. The growth of the column as a random function of the time is defined as follows. A transition \((\omega_i,\omega_{i+1})\to(\omega_i-1,\omega_{i+1}+1)\) happens with rate \(r(\omega_i,\omega_{i+1})\), where \(r\colon I^2\to\mathbb R\) is a function, satisfying the following conditions: (1) the behavior at the boundaries of \(I\) is such that the configuration stays in \(I^{\mathbb Z}\), (2) \(r\) is monotonous, more precisely, \(r(z+1,y)\geq r(y,z)\) and \(r(y,z+1)\leq r(y,z)\) for all \(y,z\in I\) (i.e., the higher the column of the neighbors is, the faster it grows), and (3) \(r\) has the form \(r(y,z)=s(y,z+1)f(y)\) for some function \(f\) and some symmetric function \(s\) (this ensures a certain product structure of the stationary measure). This model contains three models that have been studied earlier (and are all nearest-neighbor models): generalized exclusion models, generalized misantrophe processes, and general deposition models.

Let \(((h_i(t))_i)_{t\geq 0}\) be the stochastic process of brick column heights, and consider, for some parameter \(V\in\mathbb R\), the quantity \(J^{(V)}(t)=h_{Vt}(t)-h_0(0)\). The first main result of the paper is the law of large numbers for \(J^{(V)}\), more precisely, the almost sure convergence of \(J^{(V)}(t)/t\) towards \(E(r)-VE(\omega)\) as \(t\to\infty\). Under a condition on a law of large numbers and the asymptotic second moments for the so-called defect tracer \(Q(t)\) (also called the second class particle), the asymptotics of the variance of \(J^{(V)}\) and a central limit theorem are found. More precisely, \(\text{Var}(J^{(V)}(t))/t\) converges towards \(|V-C(\theta)|\text{Var}_\theta(\omega^0)\) where \(C(\theta)\) is the (assumed) limit of \(Q(t)/t\) under initial distribution of the system, indexed by \(\theta\), and \(\text{Var}_\theta\) is the variance under this initial distribution, and the appropriately centered and normalized \(J^{(V)}(t)\) converges in distribution to a standard normal distribution. The value of \(C(\theta)\) is identified for some special cases, and the case \(C(\theta)=0\) is heuristically discussed.

The model is as follows. Let \(I\) be a (bounded or unbounded) interval of integers, and consider the following random process on the state space \(\Omega=I^{ \mathbb Z}\). Assign to each edge \((i,i+1)\) the height, \(h_i\), of a column built of bricks which lie on the edge. The configuration \(\omega=(h_{i-1}-h_i)_{i\in\mathbb Z}\) is the collection of negative gradients of the heights of the ‘wall\'. The growth of the column as a random function of the time is defined as follows. A transition \((\omega_i,\omega_{i+1})\to(\omega_i-1,\omega_{i+1}+1)\) happens with rate \(r(\omega_i,\omega_{i+1})\), where \(r\colon I^2\to\mathbb R\) is a function, satisfying the following conditions: (1) the behavior at the boundaries of \(I\) is such that the configuration stays in \(I^{\mathbb Z}\), (2) \(r\) is monotonous, more precisely, \(r(z+1,y)\geq r(y,z)\) and \(r(y,z+1)\leq r(y,z)\) for all \(y,z\in I\) (i.e., the higher the column of the neighbors is, the faster it grows), and (3) \(r\) has the form \(r(y,z)=s(y,z+1)f(y)\) for some function \(f\) and some symmetric function \(s\) (this ensures a certain product structure of the stationary measure). This model contains three models that have been studied earlier (and are all nearest-neighbor models): generalized exclusion models, generalized misantrophe processes, and general deposition models.

Let \(((h_i(t))_i)_{t\geq 0}\) be the stochastic process of brick column heights, and consider, for some parameter \(V\in\mathbb R\), the quantity \(J^{(V)}(t)=h_{Vt}(t)-h_0(0)\). The first main result of the paper is the law of large numbers for \(J^{(V)}\), more precisely, the almost sure convergence of \(J^{(V)}(t)/t\) towards \(E(r)-VE(\omega)\) as \(t\to\infty\). Under a condition on a law of large numbers and the asymptotic second moments for the so-called defect tracer \(Q(t)\) (also called the second class particle), the asymptotics of the variance of \(J^{(V)}\) and a central limit theorem are found. More precisely, \(\text{Var}(J^{(V)}(t))/t\) converges towards \(|V-C(\theta)|\text{Var}_\theta(\omega^0)\) where \(C(\theta)\) is the (assumed) limit of \(Q(t)/t\) under initial distribution of the system, indexed by \(\theta\), and \(\text{Var}_\theta\) is the variance under this initial distribution, and the appropriately centered and normalized \(J^{(V)}(t)\) converges in distribution to a standard normal distribution. The value of \(C(\theta)\) is identified for some special cases, and the case \(C(\theta)=0\) is heuristically discussed.

Reviewer: Wolfgang König (Berlin)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |