Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits

TL;DR

This paper analyzes directed paths in a specific model of random planar maps, revealing their scaling limits and dimensions, and connects these findings to Liouville quantum gravity without relying on continuum theory.

Contribution

It provides exact exponents and scaling limits for directed distances in bipolar-oriented triangulations, using a bijection approach without continuum theory.

Findings

Directed path lengths scale as n^{3/4} and n^{3/8} in typical subsets.

Busemann functions converge to stable Lévy processes in the scaling limit.

Results suggest connections to Liouville quantum gravity metrics.

Abstract

We study longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge. We construct the Busemann function which measures directed distance to $\infty$ along a natural interface in the UIBOT. We show that in the case of longest (resp.\ shortest) directed paths, this Busemann function converges in the scaling limit to a $2/3$-stable L\'evy process (resp.\ a $4/3$-stable L\'evy process). We also prove up-to-constants bounds for directed distances in finite bipolar-oriented triangulations sampled from a Boltzmann distribution, and for size-$n$ cells in the UIBOT. These bounds imply that in a typical subset of the UIBOT with $n$ edges, longest directed path lengths are of order $n^{3/4}$ and shortest directed path lengths are of order $n^{3/8}$. These results give the scaling dimensions for discretizations of the (hypothetical) $\sqrt{4/3}$-directed Liouville quantum gravity metrics. The main external input in our proof is the bijection of Kenyon-Miller-Sheffield-Wilson (2015). We do not use any continuum theory. We expect that our techniques can also be applied to prove similar results for directed distances in other random planar map models and for longest increasing subsequences in pattern-avoiding permutations.