# Generalizations of TASEP in Discrete and Continuous Inhomogeneous Space

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## Abstract

We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson–Schensted–Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams. For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar–Parisi–Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy–Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions. A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last-passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results. The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.

## Notes

### Acknowledgements

We are grateful to Guillaume Barraquand, Riddhipratim Basu, Alexei Borodin, Eric Cator, Francis Comets, Ivan Corwin, Patrik Ferrari, Vadim Gorin, Pavel Krapivsky, Alexander Povolotsky, Timo Seppäläinen, and Jon Warren for helpful discussions. A part of the work was completed when the authors attended the 2017 IAS PCMI Summer Session on Random Matrices, and we are grateful to the organizers for the hospitality and support. AK was partially supported by the NSF Grant DMS-1704186. LP was partially supported by the NSF grant DMS-1664617.

## Supplementary material

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