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5. Colored particles on the ring


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      Problem 5.1.

      Can one derive an analogue of the TASEP speed process [MR2857238], [MR4576839] for colored particle systems (TASEP or ASEP, for starters) evolving on a ring?


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          Problem 5.2.

          Does a color-position symmetry exist in particle systems on a ring? (Model particular case: Half-open systems?) For models on the full space or in half-space geometries, such symmetries have been crucial for deriving exact formulas and understanding integrable structures. There are many references on color-position symmetry and the related property of shift-invariance, e.g., [MR4260463], [arXiv:2003.02730], [MR4317703], [MR4397419], [MR4385122].


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              Problem 5.3.

              Can the dynamics of colored particles on a ring be recast in terms of an (affine) Hecke algebra structure?


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                  Problem 5.4.

                  Is there an analogue of color-position symmetry within the framework of LPP models? Recent works on “hidden invariance” cited above hint at deep invariance properties that might extend to colored systems.


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                      Problem 5.5.

                      Within the known stationary measures of ASEP, can we investigate correlations? Consider the asymmetric simple exclusion process (ASEP) on a large ring with two particle colors, with particle densities $\rho_1$ and $\rho_2$. Define $\eta_i(x)$ as the indicator for the presence of a particle of color $i$ at site $x$ under stationarity. A key question is to understand how the covariance between particle occupations decays with increasing spatial separation. When densities are close to $1/2$, one could leverage the convergence to Brownian motions with drift and known stationary horizon properties to conjecture explicit forms for the correlation decay. For general densities, connections with known ASEP or TASEP analogues, such as speed processes or hidden invariances, may provide valuable insights.

                          Cite this as: AimPL: All roads to the KPZ universality class, available at http://aimpl.org/roadtokpz.