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Intermittency on catalysts: voter model. (English) Zbl 1223.60080

Let \(u:\mathbb Z^d\times[0,\infty)\to\mathbb R\) be the solution of the parabolic Anderson equation \(\partial u/\partial t=\kappa\Delta u+\gamma\xi u\), where \(\kappa\geq0\) is a parameter, \(\Delta\) is the discrete Laplacian on \(\mathbb Z^d\), \(\gamma>0\) is a constant and \(\xi:\mathbb Z^d\times[0,\infty)\to\mathbb R\) is a space-time dependent random medium. Assume the constant initial condition \(u(x,0)=1\) for all \(x\). The (annealed) Lyapunov exponents are the exponential growth rates of the \(p\)th moments; that is (if the limit exists)
\[ \lambda_p:=\lim_{t\to\infty}(pt)^{-1}\log\mathbb{E}[u(0,t)^p]. \]
The model is said to be intermittent if \(\lambda_1<\lambda_2<\dots\).
The model has been studied for random media \(\xi\) of different complexity before (see earlier papers by the authors [“Intermittency on catalysts: three-dimensional simple symmetric exclusion”, Electron. J. Probab. 14, 2091–2129 (2009; Zbl 1192.60106); “Intermittency on catalysts”, in: J. Blath (ed.) et al., Trends in stochastic analysis. Papers dedicated to Professor Heinrich von Weiszäcker on the occasion of his 60th birthday. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 353, 235–248 (2009; Zbl 1172.82331); “Intermittency on catalysts: symmetric exclusion”, Electron. J. Probab. 12, 516–573 (2007; Zbl 1129.60061)], and the first two authors [“Intermittency in a catalytic random medium”, Ann. Probab. 34, No. 6, 2219–2287 (2006; Zbl 1117.60065)]).
Here, the authors focus on a medium whose random dynamics is not reversible, namely the voter model with transition kernel \(p(\cdot,\cdot)\) on \(\mathbb Z^d\). If the initial condition is the Bernoulli measure \(\nu_\rho\) with parameter \(\rho\in(0,1)\), then the time \(t\) distribution (denoted by \(\nu_\rho S_t\)) converges to some equilibrium \(\nu_\rho S_\infty\) as \(t\to\infty\).
In their first theorem, the authors show that the Lyapunov exponent \(\lambda_p=\lambda_p(\kappa)\) exists if the initial distribution of the voter model is \(\nu_\rho S_t\) for some \(t\in[0,\infty]\) and does not depend on \(t\). They show that \(\lambda_p(\cdot)\) is globally Lipschitz for \(\kappa\in[\varepsilon,\infty)\) for any \(\varepsilon>0\). Furthermore, \(\lambda_p(\kappa)>\rho\gamma\) for all \(\kappa\geq0\). This means that the exponential growth rate is strictly larger than that of a homogeneous medium with intensity \(\rho\).
For a recurrent interaction kernel, the voter model converges to the trivial equilibrium which is a mixture of the constant configurations \(\xi\equiv0\) and \(\xi\equiv1\). For those however, it is clear that \(\lambda_p(\kappa)=\gamma\) for all \(p\) and \(\kappa\). The authors show that this is true in more generality: \(\lambda_p(\kappa)=\gamma\) if \(d\leq 4\) and \(p(\cdot,\cdot)\) has zero mean and finite variance.
In the case \(d\geq5\), the behaviour is more intricate. In fact, for large \(\kappa\), the exponents approach those of the homogeneous medium with intensity \(\rho\); that is, \(\lambda_p(\kappa)\to\rho\gamma\) as \(\kappa\to\infty\). If in addition \(p(\cdot,\cdot)\) has zero mean and finite variance, then there exists a \(\kappa_0>0\) such that, for all \(\kappa<\kappa_0\), the system is intermittent.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60H25 Random operators and equations (aspects of stochastic analysis)
60F10 Large deviations
35B40 Asymptotic behavior of solutions to PDEs
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References:

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