Convergence of exclusion processes and the KPZ equation to the KPZ fixed point.(English)Zbl 1520.60063
The Kardar-Parisi-Zhang (KPZ) equation in one-dimension is \[\partial_{t}h=\lambda (\partial_{x}h)+\nu \partial_{x}^{2}h+\sqrt{D}\xi,\]where \(\xi(x,t)\) is the white noise on \(x\in\mathbb{R}\) and \(t\geq 0\). This model encapsulates a type of universality that has been gradually unveiled, and it is called the KPZ universality. This phenomenon had been studied for several models, in particular for some particle systems; one well known is the so-called TASEP (totally asymmetric simple exclusion process).
The TASEP model are particles in one-dimension such that their interaction (they resemble cars in one street lane pushing each other to move forward) creates a variable to growth, and the fluctation of such variable (after some scaling) converges to an object which may be calledKPZ fixed point.
In this article the authors prove that the solution of the KPZ equation, a real-valued function \(h(x,t)\), is such that \[\lim_{\delta\to 0}\delta h(x/\delta^{2}, t/\delta^{3} ) \]is indeed the KPZ fixed point; this is called the \(1:2:3\) scaling, which arises in this type of limit theorems. Of course, they use the fact that for such an equation it was previously proved the existence of its solution.
Moreover, the authors also extend the convergence of other particle systems to the KPZ fixed point, such as the ASEP (asymmetric simple exclusion process) and the AEP (finite range asymmetric exclusion processes), which are more general versions of the TASEP model. A delicate issue is about the initial conditions of the models; it becomes difficult to prove the results for very general initial data, nevertheless the authors cover several situations. One important tool to carry out the proofs are the so-called Dirichlet forms to deal with the generator of the Markov processes.
The TASEP model are particles in one-dimension such that their interaction (they resemble cars in one street lane pushing each other to move forward) creates a variable to growth, and the fluctation of such variable (after some scaling) converges to an object which may be calledKPZ fixed point.
In this article the authors prove that the solution of the KPZ equation, a real-valued function \(h(x,t)\), is such that \[\lim_{\delta\to 0}\delta h(x/\delta^{2}, t/\delta^{3} ) \]is indeed the KPZ fixed point; this is called the \(1:2:3\) scaling, which arises in this type of limit theorems. Of course, they use the fact that for such an equation it was previously proved the existence of its solution.
Moreover, the authors also extend the convergence of other particle systems to the KPZ fixed point, such as the ASEP (asymmetric simple exclusion process) and the AEP (finite range asymmetric exclusion processes), which are more general versions of the TASEP model. A delicate issue is about the initial conditions of the models; it becomes difficult to prove the results for very general initial data, nevertheless the authors cover several situations. One important tool to carry out the proofs are the so-called Dirichlet forms to deal with the generator of the Markov processes.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 82C24 | Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics |
Cite
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