Nicolas Curien

Percolations on random maps I: Half-plane models

We study Bernoulli percolations on random lattices of the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process [5] of these random lattices we prove a surprisingly simple universal formula for the critical threshold for bond and face percolations on these graphs. Our techniques also permit us to compute off-critical and critical exponents related to percolation clusters such as the volume and the perimeter.

Scaling limits for the peeling process on random maps

We study the scaling limit of the volume and perimeter of the discovered regions in the Markovian explorations known as peeling processes for infinite random planar maps such as the uniform infinite planar triangulation (UIPT) or quadrangulation (UIPQ). In particular, our results apply to the metric exploration or peeling by layers algorithm, where the discovered regions are (almost) completed balls, or hulls, centered at the root vertex. The scaling limits of the perimeter and volume of hulls can be expressed in terms of the hull process of the Brownian plane studied in our previous work. Other applications include the metric exploration of the dual graph of our infinite random lattices, and first-passage percolation with exponential edge weights on the dual graph, also known as the Eden model or uniform peeling.

On limits of graphs sphere packed in Euclidean space and applications

The core of this note is the observation that links between circle packings of graphs and potential theory developed in Benjamini and Schramm (2001) [4] and He and Schramm (1995) [11] can be extended to higher dimensions. In particular, it is shown that every limit of finite graphs sphere packed in R^d with a uniformly chosen root is d-parabolic. We then derive a few geometric corollaries. For example, every infinite graph packed in R^d has either strictly positive isoperimetric Cheeger constant or admits arbitrarily large finite sets W with boundary size which satisfies |@?W|=<|W|^d^-^1^d^+^o^(^1^). Some open problems and conjectures are gathered at the end.

Uniform infinite planar quadrangulations with a boundary

We introduce and study the uniform infinite planar quadrangulation UIPQ with a boundary via an extension of the construction of Curien et al. Curien et al., Lat Am J Probab Math Stat 49 2013 340-373. We then relate this object to its simple boundary analog using a pruning procedure. This enables us to study the aperture of these maps, that is, the maximal graph distance between two points on the boundary, which in turn sheds new light on the geometry of the UIPQ. In particular we prove that the self-avoiding walk on the UIPQ is at most diffusive.

The Brownian cactus I. Scaling limits of discrete cactuses

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space

PARTIAL MATCH QUERIES IN TWO-DIMENSIONAL QUADTREES: A PROBABILISTIC APPROACH

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The Markovian hyperbolic triangulation

, one can associate an

Geometry of infinite planar maps with high degrees

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Martingales in self-similar growth-fragmentations and their connections with random planar maps

-tree called the continuous cactus of

The harmonic measure of balls in random trees

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