Universality of slow decorrelation in KPZ growth.

*(English. French summary)*Zbl 1247.82041Kardar, Parisi and Zhang (KPZ) proposed on physical grounds that a wide variety of irreversible stochastically growing interfaces should be governed by a single stochastic PDE – this is done by the KPZ equation \(h_t(X, T)\). Since then, a lot of work has been done to make mathematical sense of this SPDE and to find solutions for a large growth time \(t\). Significant progress has been made towards understanding this equation in the one-dimensional \(d = 1\) case. However, most of the rigorous works done in studying the statistics associated with some other cases have dealt with the spatial process (obtained as the asymptotic statistics of \(h_t(X, T = 1)\) as a process in \(X\), as \(t\rightarrow\infty\)) and not on how the spatial process evolves with \(T\).

It is commonly accepted that computations of exact statistics require a level of solvability – only certain solvable discrete growth models or polymer models in the KPZ universality class can be taken into account. Examples are the partially/totally asymmetric simple exclusion process (P/TASEP), last passage percolation (LPP) with exponential or geometric weights, the corner growth model, the polynuclear growth (PNG) model.

For those models rigorous spatial fluctuation results have been proved. Some progress was made on analyzing the solution of the KPZ equation itself, but it still relied on the approximation by a solvable discrete model.

The authors introduce a generalized LPP model which encompasses several KPZ class models. They give sufficient conditions under which such LPP models display slow decorrelation. These conditions (the existence of a limit shape and a one-point fluctuation result) are very elementary and hold for all the solvable models mentioned above, and are believed to hold for all KPZ class models.

This paper is divided into three sections. In Section 2, the general framework of LPP models is given with a set of criteria for slow decorrelation (Theorem 2.1). This theorem is applied to various models in the KPZ class, which can be related in some way with an LPP model, i.e., the corner growth model, point to point and point to line LPP models, TASEP, PASEP (which requires a slightly different argument since it cannot be directly mapped to a LPP problem) and the PNG models – details are provided in Section 3.

It is commonly accepted that computations of exact statistics require a level of solvability – only certain solvable discrete growth models or polymer models in the KPZ universality class can be taken into account. Examples are the partially/totally asymmetric simple exclusion process (P/TASEP), last passage percolation (LPP) with exponential or geometric weights, the corner growth model, the polynuclear growth (PNG) model.

For those models rigorous spatial fluctuation results have been proved. Some progress was made on analyzing the solution of the KPZ equation itself, but it still relied on the approximation by a solvable discrete model.

The authors introduce a generalized LPP model which encompasses several KPZ class models. They give sufficient conditions under which such LPP models display slow decorrelation. These conditions (the existence of a limit shape and a one-point fluctuation result) are very elementary and hold for all the solvable models mentioned above, and are believed to hold for all KPZ class models.

This paper is divided into three sections. In Section 2, the general framework of LPP models is given with a set of criteria for slow decorrelation (Theorem 2.1). This theorem is applied to various models in the KPZ class, which can be related in some way with an LPP model, i.e., the corner growth model, point to point and point to line LPP models, TASEP, PASEP (which requires a slightly different argument since it cannot be directly mapped to a LPP problem) and the PNG models – details are provided in Section 3.

Reviewer: Dominik Strzałka (Rzeszów)

##### MSC:

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

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

60B20 | Random matrices (probabilistic aspects) |

82B43 | Percolation |

##### Keywords:

asymmetric simple exclusion process; interacting particle systems; last passage percolation; directed polymers; KPZ
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\textit{I. Corwin} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 1, 134--150 (2012; Zbl 1247.82041)

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