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Eleanor Archer

Publications and source records attributed to Eleanor Archer.

15 recordsLinked to original sources
arXiv2025

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 $\alpha$-stable offspring tails for some $\alpha \in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $\alpha$-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↗
arXiv2024

Stable quadrangulations and stable spheres

We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaym\'e-Galton-Watson trees with stably-decaying offspring tails with an exponent $\alpha$ in (1, 2). We show that these quadrangulations admit subsequential scaling limits wich all have Hausdorff dimension $\frac{2\alpha}{\alpha-1}$ almost surely. We conjecture that the limits are unique and spherical, and we introduce a candidate for the limit that we call the $\alpha$-stable sphere. In addition, we conduct a detailed study of volume fluctuations around typical points in the limiting maps, and show that the fluctuations share similar characteristics with those of stable trees.

math.PR↗
arXiv2024

Catalan percolation

In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation. We prove that the critical $p_{\mathrm c}$ is strictly between that of oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$. Our main result shows that an enhanced oriented percolation model, with non-decaying infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hil\'ario, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman--Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hil\'ario, S\'a, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira.

math.PR↗
arXiv2024

Some properties of stable snakes

We prove some technical results relating to the Brownian snake on a stable L\'evy tree. This includes some estimates on the range of the snake, estimates on its occupation measure around its minimum and also a proof of the fact that the snake and the height function of the associated tree have no common increase points.

math.PR↗
arXiv2024

Random choice spanning trees

In this paper we introduce a new model of random spanning trees that we call choice spanning trees, constructed from so-called choice random walks. These are random walks for which each step is chosen from a subset of random options, according to some pre-defined rule. The choice spanning trees are constructed by running a choice modified version of Wilson's algorithm or the Aldous-Broder algorithm on the complete graph. We show that the scaling limits of these choice spanning trees are slight variants of random aggregation trees previously considered by Curien and Haas (2017). Moreover, we show that the loop-erasure of a choice random walk run on the complete graph converges after rescaling to a generalized Rayleigh process, extending a result of Evans, Pitman and Winter (2006). These are all natural extensions of similar results for uniform spanning trees.

math.PR↗
arXiv2023

Quenched critical percolation on Galton-Watson trees

We consider critical percolation on a supercritical Galton-Watson tree. We show that, when the offspring distribution is in the domain of attraction of an $\alpha$-stable law for some $\alpha \in (1,2)$, or has finite variance, several annealed properties also hold in a quenched setting. In particular, the following properties hold for the critical root cluster on almost every realisation of the tree: (1) the rescaled survival probabilities converge; (2) the Yaglom limit or its stable analogue hold - in particular, conditioned on survival, the number of vertices at generation $n$ that are connected to the root cluster rescale to a certain (explicit) random variable; (3) conditioned on initial survival, the sequence of generation sizes in the root cluster rescales to a continuous-state branching process. This strengthens some earlier results of Michelen (2019) who proved (1) and (2) in the case where the initial tree has an offspring distribution with all moments finite.

math.PR↗
arXiv2023

The union of independent USFs on $\mathbb{Z}^d$ is transient

We show that the union of two or more independent uniform spanning forests (USF) on $\mathbb{Z}^d$ with $d\geq 3$ almost surely forms a connected transient graph. In fact, this also holds when taking the union of a deterministic everywhere percolating set and an independent $\epsilon$-Bernoulli percolation on a single USF sample.

math.PR↗
arXiv2023

The GHP scaling limit of uniform spanning trees of dense graphs

We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the uniform spanning trees of the Erdös-Rényi sequence $(G(n,p))_{n \geq 1}$ for any fixed $p \in (0,1]$, and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law of a simple random walk.

math.PR↗
arXiv2022

Random walks on decorated Galton-Watson trees

In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the tree structure. We assume that there exist constants $d, R \geq 1, v < \infty$ such that the diameter, effective resistance across and volume of $G_n$ respectively grow like $n^{\frac{1}{d}}, n^{\frac{1}{R}}, n^v$ as $n \to \infty$. We also assume that the underlying Galton-Watson tree is critical with offspring tails decaying like $cx^{-α}$ for some constant $c>0$ and some $α\in (1,2)$. We establish the fractal dimension, spectral dimension, walk dimension and simple random walk displacement exponent for the resulting metric space as functions of $α, d, R$ and $v$, along with bounds on the fluctuations of these quantities.

math.PR↗
arXiv2022

The GHP scaling limit of uniform spanning trees in high dimensions

We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$ with $d>4$, the hypercube $\{0,1\}^n$, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.

math.PR↗
arXiv2022

Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier results on $\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.

math.PR↗
arXiv2021

Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees

We prove an invariance principle for linearly edge reinforced random walks on $\gamma$-stable critical Galton-Watson trees, where $\gamma \in (1,2]$ and where the edge joining $x$ to its parent has rescaled initial weight $d(\rho, x)^{\alpha}$ for some $\alpha \leq 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.

math.PR↗
arXiv2020

Brownian motion on stable looptrees

In this article, we introduce Brownian motion on stable looptrees using resistance techniques. We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of $r^α$ and $t^{\frac{-α}{α+ 1}}$ respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

math.PR↗
arXiv2020

Infinite stable looptrees

We give a construction of an infinite stable looptree, which we denote by $\mathcal{L}_α^{\infty}$, and prove that it arises both as a local limit of the compact stable looptrees of Curien and Kortchemski (2015), and as a scaling limit of the infinite discrete looptrees of Richier (2017) and Björnberg and Stefánsson (2015). As a consequence, we are able to prove various convergence results for volumes of small balls in compact stable looptrees, explored more deeply in a companion paper. We also establish the spectral dimension of $\mathcal{L}_α^{\infty}$, and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on $\mathcal{L}_α^{\infty}$ arises as a scaling limit of random walks on discrete looptrees, and as a local limit of Brownian motion on compact stable looptrees, which has similar consequences for the limit of the heat kernel.

math.PR↗