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Stationary half-space geometric last passage percolation

Jiyue Zeng

Abstract

We consider the half-space geometric Last Passage Percolation model starting with stationary measures. We obtain exact formulas for LPP value along the diagonal $(N,N)$ across the entire phase diagram. We also obtain the limits of these distributions under critical scaling which should yield the one-point distribution of the half-space KPZ fixed point starting from stationary initial conditions.

Stationary half-space geometric last passage percolation

Abstract

We consider the half-space geometric Last Passage Percolation model starting with stationary measures. We obtain exact formulas for LPP value along the diagonal across the entire phase diagram. We also obtain the limits of these distributions under critical scaling which should yield the one-point distribution of the half-space KPZ fixed point starting from stationary initial conditions.

Paper Structure

This paper contains 45 sections, 49 theorems, 324 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Half-space geometric last passage percolation model with initial condition
  3. Phase Diagram
  4. Main results
  5. Our strategy
  6. Outline.
  7. Acknowledgements.
  8. Basic notations
  9. Definition of distribution functions for High density phase
  10. Finite time diagonal distribution for High density phase
  11. Asymptotic limits under High density phase
  12. Definition of distribution functions for Maximal current phase
  13. Finite time diagonal distribution for Maximal current phase
  14. Asymptotic limits under Maximal current phase
  15. Definitions of distribution functions for Low density phase
  16. ...and 30 more sections

Key Result

Proposition 1.3

For any fixed $\sqrt{q} \in (0,1),$$s\in (\sqrt{q},1),$$r \in (0,1/s],$ the process $(\mathcal{I}_{r,s}(x))_{x\in\mathbb{Z}_{\geq 0}}$ is stationary for the half-space geometric LPP model recurrence.

Figures (10)

  • Figure 1: This picture describes the phase diagram of the geometric LPP model given boundary parameter and drift parameter. Depending on where $(-\log(r), \log(s))$ lies in the diagram, the process $G(\cdot,N) - G(N,N)$ converges to one of the three spatial processes, $\mathcal{I}_{r,s},$$\mathcal{I}_{r,1/r},$ or $\mathcal{I}_{r,1},$ which is claimed in Conjecture \ref{['conjecture']}. On the full line $r=s$ and the half line $rs=1, r\geq 1, s\leq 1$, the process converges to the Geom$(\sqrt{q}/r)$ random walk. MC, HD, and LD represent Maximal current (green), High density (blue), and Low density (red) phases. We characterize the distribution of $G(N,N)$ in the High density phase, i.e., $F_{r,s}^{HD}$, and on the two boundaries where HD meets the other phases: along the MC boundary we obtain $F_{r,1}^{MC}$, and along the LD boundary we obtain $F_{r}^{LD}$. In particular, we have $F_{r}^{LD}$ on the line $rs=1$, $s<1$ in HD, which will be explained further in \ref{['degenerate1']}.
  • Figure 2: Geometric LPP model as described in \ref{['eq:stat_wts']}.
  • Figure 3: Geometric LPP model as described in \ref{['one-paramStatModel']}.
  • Figure 4: Geometric LPP model as described in \ref{['approxModel']}.
  • Figure 5: Dark path for $z$ and grey path for $w$
  • ...and 5 more figures

Theorems & Definitions (115)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 4.1
  • ...and 105 more