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Non-intersecting Path Constructions for TASEP with Inhomogeneous Rates and the KPZ Fixed Point.

Elia Bisi1, Yuchen Liao2, Axel Saenz3

  • 1Institut für Stochastik und Wirtschaftsmathematik, Technische Universität Wien, E 105-07, Wiedner Hauptstraße 8-10, 1040 Wien, Austria.

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Summary
This summary is machine-generated.

We introduce a new method for analyzing the totally asymmetric simple exclusion process (TASEP) with complex particle and time-dependent jump rates. This approach uses combinatorics and lattice path ensembles to derive exact results for particle distributions.

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Area of Science:

  • Probability Theory
  • Statistical Mechanics
  • Combinatorics

Background:

  • The totally asymmetric simple exclusion process (TASEP) is a fundamental model in statistical mechanics.
  • Analyzing TASEP with non-uniform, time-dependent rates presents significant mathematical challenges.
  • Existing methods often struggle with the complexity introduced by inhomogeneous parameters.

Purpose of the Study:

  • To develop a novel framework for studying discrete-time TASEP with particle-dependent and time-inhomogeneous jump rates.
  • To express the system's transition kernel using combinatorial objects and determinantal point processes.
  • To derive an exact formula for the joint distribution of particle positions.

Main Methods:

  • Utilizing the combinatorics of the Robinson-Schensted-Knuth correspondence.
  • Employing intertwining relations to connect the particle system to lattice path ensembles.
  • Expressing the correlation kernel via a boundary-value problem for a discrete heat equation.

Main Results:

  • The transition kernel is represented as ensembles of weighted, non-intersecting lattice paths.
  • The joint particle distribution is formulated as a Fredholm determinant.
  • The correlation kernel is derived in terms of random walk hitting probabilities, generalizing previous results.

Conclusions:

  • The developed method provides a powerful tool for analyzing complex TASEP models.
  • The solution for the fully inhomogeneous case reveals a finer structure compared to homogeneous TASEP.
  • This work extends the understanding of interacting particle systems with realistic, non-uniform dynamics.