1. Two-dimensional particle systems and continuous growth models
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2-dimensional ASEP
Problem 1.1.
- Study the two-dimensional asymmetric simple exclusion process (ASEP) on the lattice $\mathbb{Z}^2$, where particles attempt hops to the four cardinal directions (N, S, E, W) with differing asymmetric rates. A key question is whether the resulting fluctuations can be linked to the two-dimensional KPZ universality class.
- An essential step is to define a suitable height function that maps particle configurations to a continuous interface. The challenge is to capture the local imbalance in particle flows and to ensure that the macroscopic limit is well-defined. One idea involves studying the exit time required for a particle to exit a fixed ball in $\mathbb{Z}^2$.
- Investigate how various initial conditions (e.g., a half-plane fully occupied by particles) influence the hydrodynamic scaling limits and the fluctuation behavior.
- A long-term goal is to rigorously derive the continuum stochastic partial differential equations (SPDEs) that govern the evolution of current fluctuations and stationary measures in the scaling limit.
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Isotropic 2d KPZ
Problem 1.2.
The isotropic two-dimensional KPZ problem remains largely open. A central challenge is to identify the fluctuation exponents and to characterize the fixed point process that would parallel the one-dimensional KPZ FP. -
Anisotropic models
Problem 1.3.
- In contrast to the isotropic case, anisotropic KPZ models [MR2485688], [MR3607467], [MR3855490] are expected to exhibit different scaling behaviors. A key question is whether these models converge to a distinct fixed point under appropriate scaling limits.
- One potential approach involves analyzing the coupling of Gaussian free fields (GFFs) with varying slopes to understand the invariant measures of the hypothetical two-dimensional anisotropic KPZ FP.
Cite this as: AimPL: All roads to the KPZ universality class, available at http://aimpl.org/roadtokpz.