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Tanguy Lions

Publications and source records attributed to Tanguy Lions.

7 recordsLinked to original sources
arXiv2026

Typical distances in high-genus triangulations

We study the distance between two uniformly chosen points on a uniform random triangulation whose genus g is proportional to the number of faces 2n. We show that the distance rescaled by log(n) converges in probability to a deterministic constant, which answers a conjecture of Budzinski, Chapuy and Louf. The proof relies on the precise study of the volume growth of the ball of radius r for r of order log(n). The main ingredients are the recent local convergence results for uniform triangulations with boundaries and the isoperimetric inequalities obtained by Budzinski and Louf.

math.PR↗
arXiv2026

Quenched scaling limit of critical percolation clusters on Galton-Watson trees

We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $α$-stable offspring tails for some $α\in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $α$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024).

math.PR↗
arXiv2026

Local limits of uniform triangulations with boundaries in high genus

We study the local limits of uniform random triangulations with boundaries in the regime where the genus is proportional to the number of faces. Budzinski and Louf proved in 2020 that when there are no boundaries, the local limits exist and are the Planar Stochastic Hyperbolic Triangulation (PSHT) introduced in PSHT. We show that when the triangulations considered have size n and boundaries with total length p that tends to infinity with n and p=o(n), the local limits around a typical boundary edge are the half-plane hyperbolic triangulations defined by Angel and Ray. This provides, for the first time, a construction of these hyperbolic half-plane triangulations as local limits of large genus triangulations. We also prove that under the condition p = o(n), the local limit when rooted on a uniformly chosen oriented edge is given by the PSHT. Contrary to the proof of Budzinski and Louf, the latter does not rely on the Goulden-Jackson recurrence relation, but only on coarse combinatorial estimates. Thus, we expect that the proof can be adapted to local limits in similar models.

math.PR↗
arXiv2026

The tight length spectrum of large-genus random hyperbolic surfaces with many cusps

Since the work of Mirzakhani and Petri on random hyperbolic surfaces of large genus, length statistics of closed geodesics have been studied extensively. We focus on the case of random hyperbolic surfaces with cusps, the number of which grows with the genus. We prove that if the number of cusps grows fast enough and we restrict attention to special geodesics that are tight, we recover upon proper normalization the same Poisson point process in the large genus limit for the length statistics. The proof relies on a recursion formula for tight Weil-Petersson volumes obtained recently by Budd and Zonneveld and on a generalization of Mirzakhani's integration formula to the tight setting.

math.PR↗
arXiv2026

Large expander subgraphs in high genus triangulations

We prove that random triangulations of high genus contain very large expander subgraphs, answering a question of Benjamini. Our approach relies on new general criteria for arbitrary graphs to contain large expander subgraphs.

math.CO↗
arXiv2024

A phase transition for the biased tree-builder random walk

We consider a recent model of random walk that recursively grows the network on which it evolves, namely the Tree Builder Random Walk (TBRW). We introduce a bias $ρ\in (0,\infty)$ towards the root, and exhibit a phase transition for transience/recurrence at a critical threshold $ρ_c =1+2\overlineν$, where $\overlineν$ is the (possibly infinite) expected number of new leaves attached to the walker's position at each step. This generalizes previously known results, which focused on the unbiased case $ρ=1$. The proofs rely on a recursive analysis of the local times of the walk at each vertex of the tree, after a given number of returns to the root. We moreover characterize the strength of the transience (law of large numbers and central limit theorem with positive speed) via standard arguments, establish recurrence at $ρ_c$, and show a condensation phenomenon in the non-critical recurrent case.

math.PR↗