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Uniform convergence to the Airy line ensemble

Duncan Dauvergne, Mihai Nica, Bálint Virág

Abstract

We show that classical integrable models of last passage percolation and the related nonintersecting random walks converge uniformly on compact sets to the Airy line ensemble. Our core approach is to show convergence of nonintersecting Bernoulli random walks in all feasible directions in the parameter space. We then use coupling arguments to extend convergence to other models.

Uniform convergence to the Airy line ensemble

Abstract

We show that classical integrable models of last passage percolation and the related nonintersecting random walks converge uniformly on compact sets to the Airy line ensemble. Our core approach is to show convergence of nonintersecting Bernoulli random walks in all feasible directions in the parameter space. We then use coupling arguments to extend convergence to other models.

Paper Structure

This paper contains 8 sections, 22 theorems, 178 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Nonintersecting Bernoulli walks and the Airy line ensemble
  3. Kernel convergence for Bernoulli walks
  4. Uniform convergence
  5. Convergence Theory
  6. The Bernoulli case
  7. The geometric environment
  8. Last passage percolation in other environments

Key Result

Theorem \oldthetheorem

Consider a sequence of last passage percolation models, indexed by $n \in \mathbb{N}$, with independent geometric random variables of mean $\beta_n^{-1} \in (0, \infty)$. Let $m_n$ be a sequence of positive integers: we will analyze last passage values (defined precisely in D:LPP-k) from the bottom- which is the deterministic approximation of the last passage value $L_{n, 1}(m)$. We now define the

Figures (7)

  • Figure 1: Realizations of differences in last passage percolation in an environment of i.i.d. geometric random variables: $P_k(t) := L_{n,{k+1}}(t) - L_{n,{k}}(t) - k + 1$. These walks are identical in distribution to $n$ random walks whose increments are geometric random variables of mean $\beta^{-1}$ that are conditioned not to intersect for all time, see Section \ref{['S:LPP-geometric']}. The arctic curve is displayed in red. Theorem \ref{['T:main-lpp']} describes the fluctuation limit.
  • Figure 2: Realizations of nonintersecting Bernoulli random walks for different parameters $\beta$ and $n$. The arctic curve is shown in red. A contour integral formula allows computation of the fluctuations around the arctic curve in Theorem \ref{['T:main-walk']}.
  • Figure 3: The contour $\mathcal{C}_w$ in Proposition \ref{['P:G1-bounds-w']} for positive times. It starts at a point $\delta - \eta$ for some small $\eta$ and stays within a circle of radius $\delta - \eta$ about the origin. Moreover, $\Re(L)$ descends proportional to $L'$ along the entire contour. We first follow a straight line emanating from the point $\delta - \eta$ and then append a circular arc about the origin. The point at which $\mathcal{C}_w$ switches from following a straight line to a circular arc is chosen so that $\mathcal{C}_w$ always stays away from the point $-\beta^{-1}$.
  • Figure 4: A sketch of possibilities for the contour $\mathcal{C}_z$ in Proposition \ref{['P:G1-bounds-z']}. The contour starts $\delta + \eta$ for some small $\eta$, stays outside of a circle of radius $\delta + \eta$ about the origin, and $\Re(L)$ ascends proportionally to $L'$ along the entire contour. It is a piecewise construction whereby that first follows a straight line from the point $\delta - \eta$. If the directional derivative of $\Re(L)$ becomes too small at some point, then we turn either left along another straight line, or right along a circle centered at $1$. One of these choices guarantees that $\Re(L)$ ascends at a fast enough rate. If $\mathcal{C}_z$ turns to the right, then $t_z < \infty$; otherwise, $t_z = \infty$.
  • Figure 5: The paths $L_{n,k}(m)$ in Definition \ref{['D:LPP-discrete']} for $n=5, m=4, k=2$. $W$ is indexed by a quadrant with the bottom left corner being $(1,1)$.
  • ...and 2 more figures

Theorems & Definitions (51)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem: Nonintersecting Bernoulli walks
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 41 more