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

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.