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Periodic KPZ fixed point with general initial conditions

Jinho Baik, Yuchen Liao, Zhipeng Liu

Abstract

We consider the periodic totally asymmetric simple exclusion process with a general initial condition that properly approximates a periodic upper-semicontinuous function. We find the large time limit of the rescaled space-time multipoint distribution of the height function in the relaxation time scale. The limiting functions form a consistent family of finite-dimensional distributions; thus, they define the periodic KPZ fixed point with a general upper-semicontinuous initial condition. The main technical novelty of the paper is a hitting expectation representation of the energy function and the characteristic function in the finite-time multipoint distribution formula obtained in arXiv:1912.10143. The representation of the characteristic function is partly inspired by the work of arXiv:1701.00018, arXiv:2509.03246, while the representation of the energy function is based on an extensive guess-and-check exploration.

Periodic KPZ fixed point with general initial conditions

Abstract

We consider the periodic totally asymmetric simple exclusion process with a general initial condition that properly approximates a periodic upper-semicontinuous function. We find the large time limit of the rescaled space-time multipoint distribution of the height function in the relaxation time scale. The limiting functions form a consistent family of finite-dimensional distributions; thus, they define the periodic KPZ fixed point with a general upper-semicontinuous initial condition. The main technical novelty of the paper is a hitting expectation representation of the energy function and the characteristic function in the finite-time multipoint distribution formula obtained in arXiv:1912.10143. The representation of the characteristic function is partly inspired by the work of arXiv:1701.00018, arXiv:2509.03246, while the representation of the energy function is based on an extensive guess-and-check exploration.
Paper Structure (32 sections, 24 theorems, 309 equations, 2 figures)

This paper contains 32 sections, 24 theorems, 309 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Further discussions about the main results
  4. Proof strategy and structure of the paper
  5. The multipoint distributions of the periodic KPZ fixed point
  6. Definition of $\mathrm{C}_\mathfrak{h}(\mathbf{z})$
  7. Definition of $\mathrm{D}_\mathfrak{h}(\mathbf{z})$
  8. Fredholm determinant definition of $\mathrm{D}_\mathfrak{h}(\mathbf{z})$
  9. Series expansion of $\mathrm{D}_\mathfrak{h}(\mathbf{z})$
  10. Periodic TASEP with general initial conditions
  11. Bethe roots and the energy and characteristic functions
  12. Probabilistic representations of the energy and characteristic functions
  13. Multipoint distributions of periodic TASEP with general initial conditions
  14. Proof of the PTASEP formulas
  15. Well-definedness of the characteristic functions
  16. ...and 17 more sections

Key Result

Theorem 1.2

Assume that $\mathfrak{h}\in\mathrm{UC}_\mathrm{p}$, and $H^{\mathrm{PTASEP}}_{L;\mathfrak{h}}(x,0)$ after rescaling converges to $\mathfrak{h}$ in the space $\mathrm{UC}_\mathrm{p}$ as $L\to \infty$ as described above. For any $m\ge 1$, and any $m$ distinct points $(\beta_\ell,\alpha_\ell,\tau_\ell for $1\leq i\leq m$. Then we have The limit $\mathbb{F}^{(\mathrm{p})}_\mathfrak{h}$ is defined by

Figures (2)

  • Figure 1: The roots of $\mathrm{e}^{-\zeta^2/2}=\mathrm{z}$ for $\mathrm{z}=0.3\mathrm{e}^{\mathrm{i}\pi/4}$. The dashed line is the level curve $|\mathrm{e}^{-\zeta^2/2}|=|\mathrm{z}|$.
  • Figure 2: Roots and level sets for $w^N(w+1)^{L-N} = z^L$ with $N=6, L=18$. Here $\mathbf{r}_0 =\frac{4^{1/3}}{3}$.

Theorems & Definitions (55)

  • Definition 1.1: Space of the initial height functions
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.5
  • proof : Proof of Theorem \ref{['thm:main2']}
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5: Shift invariane of $\mathrm{C}_\mathfrak{h}(\mathbf{z})$
  • ...and 45 more