[Paper Review] Height fluctuations of stationary TASEP on a ring in relaxation time scale

Zhipeng Liu|arXiv (Cornell University)|Oct 14, 2016

Random Matrices and Applications12 references3 citations

TL;DR

This paper establishes a rigorous limit theorem for height fluctuations in the stationary totally asymmetric simple exclusion process (TASEP) on a ring at the relaxation time scale, where time scales as $ t \sim L^{3/2} $. It derives the limiting distribution $ F_U(x; \tau, \gamma) $ of the height function along the characteristic line, revealing a crossover between KPZ and Gaussian fluctuations with explicit dependence on time $ \tau $ and spatial location $ \gamma $, generalizing prior non-rigorous results to a broader class of initial conditions including non-half-filled systems.

ABSTRACT

We consider the totally asymmetric simple exclusion process on a ring with stationary initial conditions. The crossover between KPZ dynamics and equilibrium dynamics occurs when time is proportional to the $3/2$ power of the ring size. We obtain the limit of the height function along the direction of the characteristic line in this time scale. The two-point covariance function in this scale is also discussed.

Motivation & Objective

• To rigorously establish the limit distribution of the height function in stationary TASEP on a ring at the relaxation time scale $ t \sim L^{3/2} $, extending prior non-rigorous results.
• To generalize previous work on step and flat initial conditions to the full class of stationary initial conditions with arbitrary particle density $ \rho \in (0,1) $, not restricted to half-filling.
• To derive the joint limit of height fluctuations along the characteristic line $ \ell = (1-2\rho)t + \gamma L $, capturing both temporal and spatial dependence in the scaling limit.
• To prove that the limiting distribution $ F_U(x;\tau,\gamma) $ is periodic in $ \gamma $ and symmetric under $ \gamma \to -\gamma $, reflecting the ring geometry and particle number conservation.

Proposed method

• The analysis uses a contour integral representation of the height function via the Karlin-McGregor formula and the Bethe ansatz, adapted to the ring geometry.
• A change of variables is applied to map the height function to a Fredholm determinant involving a kernel $ K_{\mathbf{z};k,\ell}^{(2)} $, which is then analyzed in the scaling limit.
• The method involves asymptotic analysis of the kernel using steepest descent techniques, with careful control of error terms in the $ L \to \infty $ limit.
• The limiting distribution $ F_U(x;\tau,\gamma) $ is defined via a Fredholm determinant and shown to satisfy periodicity $ F_U(x;\tau,\gamma) = F_U(x;\tau,\gamma+1) $ and symmetry $ F_U(x;\tau,\gamma) = F_U(x;\tau,-\gamma) $.
• The proof leverages a comparison with the infinite-line stationary TASEP and uses a coupling argument with a reference measure to control fluctuations.
• A key technical step involves bounding the probability of rare events via exponential decay estimates on the kernel and height function, using geometric and analytic estimates on the contour.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the height function in stationary TASEP on a ring when time scales as $ L^{3/2} $, and how does it depend on both time and spatial location?
• RQ2 How does the crossover between KPZ and Gaussian fluctuations manifest in the height function under general stationary initial conditions (not restricted to half-filling)?
• RQ3 What is the role of the ring topology in shaping the limiting distribution, particularly in terms of periodicity and symmetry?
• RQ4 Can the non-rigorous limiting distribution obtained by Prolhac for the half-filled case be rigorously justified and extended to general densities?
• RQ5 How does the two-parameter scaling $ \tau $ (time) and $ \gamma $ (spatial shift) capture the full structure of height fluctuations in the relaxation time scale?

Key findings

• The limiting distribution of the height function along the characteristic line is given by $ F_U(x; \tau, \gamma) $, which depends on both time $ \tau $ and spatial location $ \gamma = 2w\tau^{2/3} $, generalizing previous results.
• The distribution $ F_U(x; \tau, \gamma) $ is periodic in $ \gamma $, reflecting the periodicity of the ring and the conservation of particle number.
• The distribution satisfies $ F_U(x; \tau, \gamma) = F_U(x; \tau, -\gamma) $, indicating symmetry under spatial reflection.
• The limit is valid for any fixed $ \rho \in (c_1, c_2) $ with $ 0 < c_1 < c_2 < 1 $, so the result holds for a broad class of particle densities, not just $ \rho = 1/2 $.
• The asymptotic analysis confirms that the height fluctuations scale as $ t_L^{1/3} $, consistent with KPZ universality, and the limiting distribution captures the crossover from KPZ to Gaussian behavior.
• The rigorous derivation confirms the consistency of the limiting distribution with earlier non-rigorous predictions by Prolhac and Baik-Liu, resolving a discrepancy in formula form through a unified framework.