Basic properties of the infinite critical-FK random map

Recurrence of the Uniform Infinite Half-Plane Map via duality of resistances

We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance between the root and the boundary of the hull of radius $r$ is at least of order $\log r$. This resistance bound is expected to be sharp, and is better than those following from previous proofs of recurrence for non bounded-degree planar maps models. Our main tools are the self-duality of uniform planar maps, a classical lemma about duality of resistances an…

Weak Lévy-Khintchine representation for weak infinite divisibility

A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$ there exists a random variable $Θ$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} Θ$, where $X'$ is an independent copy of $X$ and $Θ$ is independent of $X$. This is equivalent (see [12]) with the condition that for all random variables $Q_1, Q_2$ there exists a random variable $Θ$ such that $$ {\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} Θ, \quad \quad \quad \quad \quad (\ast) $$ where ${\bf X}, {\bf X}', Q_1, Q_2, Θ$ are independent. In this paper we define weak generalized convolution of measures def…

Statistical properties of coupled expanding maps on a lattice with general infinite range couplings and Hölder densities

We continue the development of transfer operator techniques for expanding maps on a lattice coupled by general interaction functions. We obtain a spectral gap for an appropriately defined transfer operator, and, as corollaries, the existence of an invariant conformal probability measure for the system, exponential decay of correlations, the central limit theorem and the almost sure invariance principle.

Random infinite squarings of rectangles

A recent preprint (arXiv:1402.2632) introduced a growth procedure for planar maps, whose almost sure limit is "the uniform infinite 3-connected planar map". A classical construction of Brooks, Smith, Stone and Tutte (1940) associates a squaring of a rectangle (i.e. a tiling of a rectangle by squares) to any to finite, edge-rooted planar map with non-separating root edge. We use this construction together with the map growth procedure to define a growing sequence of squarings of rectangles. We prove the sequence of squarings converges to an almost sure limit: a random infinite squaring of a fin…

Scaling of Percolation on Infinite Planar Maps, I

We consider several aspects of the scaling limit of percolation on random planar triangulations, both finite and infinite. The equivalents for random maps of Cardy's formula for the limit under scaling of various crossing probabilities are given. The limit probabilities are expressed in terms of simple events regarding Airy-Levy processes. Some explicit formulas for limit probabilities follow from this relation by applying known results on stable processes. Conversely, natural symmetries of the random maps imply identities concerning the Airy-Levy processes.

关于无限阶梯曲面上一族拟伪阿诺索夫映射

On a family of pseudo-Anosov-like maps on the infinite ladder surface

研究无限阶梯曲面\(\mathcal{L}\)上“拟伪阿诺索夫类”映射，通过\(S_g\)上彭纳型拟伪阿诺索夫映射在覆盖\(q_g\)下的提升得到，证明其拓扑传递、混合且支持零递归动力学，还给出具体例子。