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Limiting one-point fluctuations of the geodesic in the directed landscape near the endpoints when the geodesic length goes to infinity

Zhipeng Liu, Chen Ma, Tejaswi Tripathi

Abstract

We consider the limiting fluctuations of the geodesic in the directed landscape, conditioning on its length going to infinity. It was shown in \cite{Liu22b,Ganguly-Hegde-Zhang23} that when the directed landscape $\mathcal{L}(0,0;0,1) = L$ becomes large, the geodesic from $(0,0)$ to $(0,1)$ lies in a strip of size $O(L^{-1/4})$ and behaves like a Brownian bridge if we zoom in the strip by a factor of $L^{1/4}$. Moreover, the length along the geodesic with respect to the directed landscape fluctuates of order $O(L^{1/4})$ and its limiting one-point distribution is Gaussian \cite{Liu22b}. In this paper, we further zoom in a smaller neighborhood of the endpoints when $\mathcal{L}(0,0;0,1) = L$ or $\mathcal{L}(0,0;0,1) \ge L$, and show that there is a critical scaling window $L^{-3/2}:L^{-1}:L^{-1/2}$ for the time, geodesic location, and geodesic length, respectively. Within this scaling window, we find a nontrivial limit of the one-point joint distribution of the geodesic location and length as $L\to\infty$. This limiting distribution, if we tune the time parameter to infinity, converges to the joint distribution of two independent Gaussian random variables, which is consistent with the results in \cite{Liu22b}. We also find a surprising connection between this limiting distribution and the one-point distribution of the upper tail field of the KPZ fixed point recently obtained in \cite{Liu-Zhang25}.

Limiting one-point fluctuations of the geodesic in the directed landscape near the endpoints when the geodesic length goes to infinity

Abstract

We consider the limiting fluctuations of the geodesic in the directed landscape, conditioning on its length going to infinity. It was shown in \cite{Liu22b,Ganguly-Hegde-Zhang23} that when the directed landscape becomes large, the geodesic from to lies in a strip of size and behaves like a Brownian bridge if we zoom in the strip by a factor of . Moreover, the length along the geodesic with respect to the directed landscape fluctuates of order and its limiting one-point distribution is Gaussian \cite{Liu22b}. In this paper, we further zoom in a smaller neighborhood of the endpoints when or , and show that there is a critical scaling window for the time, geodesic location, and geodesic length, respectively. Within this scaling window, we find a nontrivial limit of the one-point joint distribution of the geodesic location and length as . This limiting distribution, if we tune the time parameter to infinity, converges to the joint distribution of two independent Gaussian random variables, which is consistent with the results in \cite{Liu22b}. We also find a surprising connection between this limiting distribution and the one-point distribution of the upper tail field of the KPZ fixed point recently obtained in \cite{Liu-Zhang25}.

Paper Structure

This paper contains 14 sections, 12 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Definition of the joint density functions $\hat{\mathrm{p}}$ and $\mathrm{p}$
  4. Relations to the KPZ fixed point with the upper tail conditioning
  5. Main idea of the proof and structure of the paper
  6. Proof of Theorem \ref{['thm:main']}
  7. The joint density function of the geodesic location and lengths
  8. Asymptotic analysis and the proof of \ref{['eq:main0']}
  9. Proof of Proposition \ref{['prop:joint_density']}
  10. Bound of the conditional geodesic location
  11. Bound of the conditional tail probabilities of the geodesic length
  12. Proof of Proposition \ref{['prop:properties_p']}
  13. Large time limit of the joint density function $\mathrm{p}$ and proof of \ref{['eq:large_time_asymptotics']}
  14. Approximation of the joint density functions $\mathrm{p}$ and $\hat{\mathrm{p}}$ when $h$ or $|x|$ goes to infinity

Key Result

Theorem 1.1

Assume $t>0$ is fixed. For any $x_1,x_2,h_1,h_2\in\mathbb{R}$ satisfying $x_1<x_2$ and $h_1<h_2$, we have and as $L\to\infty$. We also have and as $L\to\infty$. Here the functions $\hat{\mathrm{p}}$ and $\mathrm{p}$ are explicitly defined in Definition def:rp.

Figures (3)

  • Figure 1: Illustration of the contours $\Gamma_\mathrm{L}$ and $\Gamma_\mathrm{R}$. Note that the $\Gamma_\mathrm{L}$ contour lies between the two points $-1$ and $0$, and the $\Gamma_\mathrm{R}$ contour lies between $0$ and $1$.
  • Figure 2: The two figures above are approximations of the joint density functions $\mathrm{p}(h,x;t=1)$ and $\hat{\mathrm{p}}(h,x;t=1)$, respectively. Due to the complexity of their formulas, only the first term was numerically computed, with the remaining terms omitted.
  • Figure 3: Illustration of the contours $C_{\mathrm{L}}^\mathrm{in}, C_{\mathrm{L}},C_{\mathrm{L}}^\mathrm{out},C_{\mathrm{R}}^\mathrm{out}, C_{\mathrm{R}},$, and $C_{\mathrm{R}}^\mathrm{in}$. Each contour on the left half plane goes from $\infty e^{-2\pi/3}$ to $\infty e^{2\pi/3}$, and each contour on the right half plane goes from $\infty e^{-\pi/3}$ to $\infty e^{\pi/3}$. Following the convention in Liu22c, we use the superscripts "in" and "out" to denote the contour positions relative to the reference point $-\infty$ or $\infty$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • proof : Proof of Theorem \ref{['thm:connection_rp_UTfield']}
  • Remark 1.9
  • ...and 11 more