Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1801.02379

How fast planar maps get swallowed by a peeling process

Nicolas Curien·Cyril Marzouk
View original source ↗Read source document ↗Export metadata ↓

Abstract

The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [Curien & Le Gall, Scaling limits for the peeling process on random maps, Ann. Inst. Henri Poincaré Probab. Stat. 53, 1 (2017), 322-357] that any peeling process will eventually discover the whole map. In this paper we study the probability that the origin is not swallowed by the peeling process until time $n$ and show it decays at least as $n^{-2c/3}$ where \[c \approx 0.12831235141783245423674486573872854933142662048339843...\] is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable Lévy process conditioned to remain positive [Chaumont, Kyprianou & Pardo, Some explicit identities associated with positive self-similar Markov processes, Stochastic Process. Appl. 119, 3 (2009), 980-1000] which appears as a scaling limit for the perimeter process. As an application we sharpen the upper bound of the sub-diffusivity exponent for random walk of [Benjamini & Curien, Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points, Geom. Funct. Anal. 23, 2 (2013), 501-531].

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nicolas Curien, Cyril Marzouk. 2018-02-28. How fast planar maps get swallowed by a peeling process. https://doi.org/10.1214/18-ecp123

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The extremal process of a cascading family of branching Brownian motion

We study the asymptotic behaviour of the extremal process of a cascading family of branching Brownian motions. This is a particle system on the real line such that each particle has a type in addition to his position. Particles of type $1$ move on the real line according to Brownian motions and branch at rate $1$ into two children of type $1$. Furthermore, at rate $α$, they give birth to children too of type $2$. Particles of type $2$ move according to standard Brownian motion and branch at rate $1$, but cannot give birth to descendants of type $1$. We obtain the asymptotic behaviour of the extremal process of particles of type $2$.

math.PR↗
arXiv2026

Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let $\{X_k\}_{k\in\mathbb Z}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal H_1$, and let $G:\mathcal H_1\to\mathcal H_2$ be a measurable map into another separable Hilbert space $\mathcal H_2$. We derive a central limit theorem for the centered normalized partial sums of the Hilbert space-valued subordinated process $\{G[X_k]\}_{k\in\mathbb Z}$. Our result holds under either of two sets of sufficient conditions, formulated in terms of the transformation $G$ and the temporal and cross-sectional dependence structure of $\{X_k\}_{k\in\mathbb Z}$. These conditions coincide in finite dimensions but lead to genuinely different phenomena in the infinite-dimensional setting. The proof relies on the recently developed Fourth Moment Theorem on Hilbert spaces, leveraging tools from the infinite-dimensional Malliavin-Stein framework. We also provide continuous-time and quantitative versions of the central limit theorem. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present, as an application of our result, a novel limit theorem in the framework of neural operators.

math.PR↗
arXiv2026

Controlled rough SDEs, pathwise stochastic control and dynamic programming principles

We study stochastic optimal control of rough stochastic differential equations (RSDEs). This is in the spirit of the pathwise control problem (Lions--Souganidis 1998, Buckdahn--Ma 2007; also Davis--Burstein 1992), with renewed interest and recent works drawing motivation from filtering, SPDEs, and reinforcement learning. Results include regularity of rough value functions, validity of a rough dynamic programming principles and new rough stability results for HJB equations, removing excessive regularity demands previously imposed by flow transformation methods. Measurable selection is used to relate RSDEs to "doubly stochastic" SDEs under conditioning. In contrast to previous works, Brownian statistics for the to-be-conditioned-on noise are not required, aligned with the "pathwise" intuition that these should not matter upon conditioning. Depending on the chosen class of admissible controls, the involved processes may also be anticipating. The resulting stochastic value functions coincide in great generality for different classes of controls. RSDE theory offers a powerful and unified perspective on this problem class.

math.PR↗