International Journal on Science and Technology

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Volume 17 Issue 3
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Study of Multipoint Distributions of Growth and Interacting Particle Models: Periodic Tasep and Its Connection to Kpz Universality

Author(s) Ms. Gadriah Jamaah Ali Madi
Country India
Abstract The Kardar-Parisi-Zhang (KPZ) universality class governs the large-scale behavior of a vast array of stochastic growth processes in 1+1 dimensions. While single-point height distributions now well characterized, the full spatiotemporal structure—captured by multipoint joint distributions—has only recently become accessible through the periodic very asymmetric simple exclusion process (TASEP). This review provides a comprehensive mathematical treatment of multipoint distributions for periodic TASEP in the critical relaxation time scale t=O (L3/2), where finite-size effects compete with KPZ dynamics. The exact present Fredholm determinant formula of Baik and Liu (2019) for the joint height distribution at arbitrary space-time points, and analyze its asymptotic limits. The resulting universal limiting distributions depend on the relaxation parameter τ=t L−3/2, interpolating between Tracy-Widom GUE (τ→0) and Gaussian (τ→∞) behavior. Recent extensions to general initial conditions, discrete-time models, and half-space geometries examined, highlighting the role of probabilistic Brownian motion representations. Establish connections to integrable differential equations, including the KP equation, and discuss numerical methods for computing these distributions. This framework provides an integrated view of the two-dimensional KPZ height field, revealing the universal structure of limiting processes with periodic boundary effects.
Keywords Integrable Probability; Interacting Particle Systems; Universal Height Field; Tracy-Widom Distribution; Bethe Ansatz
Field Mathematics
Published In Volume 17, Issue 3, July-September 2026
Published On 2026-08-02
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Crossref DOI prefix of IJSAT is 10.71097/IJSAT

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