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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