The UIPQ seen from a point at infinity along its geodesic ray

TL;DR

This paper investigates the local limits of the uniform infinite quadrangulation of the plane when re-rooted along a geodesic ray, revealing a new limit quadrangulation obtained by gluing along the geodesic.

Contribution

It identifies the limit quadrangulation obtained by rerooting the UIPQ at points along a geodesic ray and analyzes its structure using extensions of the Schaeffer correspondence.

Findings

The local limit of the re-rooted quadrangulations is characterized.

The structure of the limit quadrangulation is obtained by gluing along the geodesic.

The approach uses extensions of the Schaeffer correspondence with trees.

Abstract

We consider the uniform infinite quadrangulation of the plane (UIPQ). Curien, M\'enard and Miermont recently established that in the UIPQ, all infinite geodesic rays originating from the root are essentially similar, in the sense that they have an infinite number of common vertices. In this work, we identify the limit quadrangulation obtained by rerooting the UIPQ at a point at infinity on one of these geodesics. More precisely, calling $v_k$ the $k$-th vertex on the "leftmost" geodesic ray originating from the root, and $Q_{\infty}^{(k)}$ the UIPQ re-rooted at $v_k$, we study the local limit of $Q_{\infty}^{(k)}$. To do this, we split the UIPQ along the geodesic ray $(v_k)_{k\geq 0}$. Using natural extensions of the Schaeffer correspondence with discrete trees, we study the quadrangulations obtained on each "side" of this geodesic ray. We finally show that the local limit of $Q_{\infty}^{(k)}$ is the quadrangulation obtained by gluing the limit quadrangulations back together.