Existence of the zero range process and a deposition model with superlinear growth rates.

*(English)*Zbl 1138.60340This paper gives a rigorous construction of the bricklayer’s process with exponential jump rates and establishes its ergodicity. This process was introduced and studied by M. Balasz in [J. Statist. Phys. 105, 511–524 (2001; Zbl 1017.82035)] and [J. Statist. Phys. 117, 77–98 (2004; Zbl 1017.82035)]. The bircklayer’s process can be described either by the height process \((\underline h(t); t\geq 0)\) or by the increment process \((\underline \omega(t); t\geq 0)\) where for \(i\in\mathbb Z\), \( \omega_i = h_{i-1} - h_i\). One can set \(h_0(0)=0\). Given a configuration \(\underline h\) in \(\Omega:=\mathbb Z^\mathbb Z\) and \(i\in \mathbb Z\), let \(\underline h^{(i)}\) be the configuration
\[
(\underline h^{(i)})_j =\begin{cases} h_j & {\text{ if}}\, j\neq i\cr h_j +1 & {\text{ if}}\, j=i\end{cases}
\]
Formally, the jump \(\underline h\to \underline h^{(i)}\) of the height process occurs independently for each site \(i\) with rate \(r(\omega_i) + r(\omega_{i+1})\). The assumptions on the rate function \(r\) are that it is strictly increasing on \(\mathbb N\) with \(\lim_{z\to \infty } r(z) =+\infty\) and it has exponential growth, that is, there is a constant \(\beta> 0\) such that \(r(z) < e^{\beta z}\) for all \(z\in\mathbb N\). The jump rates for negative \(z\) are determined by \(r(z)r(1-z) = 1\). Under these conditions, there is a family of i.i.d. invariant distributions which are ergodic and extremal. It is shown that the appropriate state space \(\tilde \Omega\subset \mathbb Z^\mathbb Z\) is the space of configurations whose asymptotic slopes are Cesarò bounded in some sense. When the initial increments \(\underline \omega\) is in \(\tilde \Omega\), then \( \underline \omega(\cdot)\) and \( \underline h (\cdot)\) are both Markov processes with cadlag paths. The height process is first constructed as a monotone non-decreasing limit of systems with finitely many sites. The construction of the infinite systems uses conditional couplings to compare processes with different initial configurations and attractivity with bounds on the probability that a large block of adjacent sites all experience a jump in a given time interval. Analytic properties of the semigroup and the generator are also obtained.

Reviewer: Daniel Boivin (Brest)

##### MSC:

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

82C22 | Interacting particle systems in time-dependent statistical mechanics |

##### Keywords:

Zero range; bricklayer’s; construction of dynamics; ergodicity of dynamics; superlinear jump rates
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\textit{M. Balázs} et al., Ann. Probab. 35, No. 4, 1201--1249 (2007; Zbl 1138.60340)

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