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Nicolas Curien

Publications and source records attributed to Nicolas Curien.

At least 73 records · Page 4Linked to original sources
arXiv2014

Random stable looptrees

We introduce a class of random compact metric spaces L(α) indexed by α\in (1,2) and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be informally be viewed as dual graphs of α-stable Lévy trees. We study their properties and prove in particular that the Hausdorff dimension of L(α) is almost surely equal to α. We also show that stable looptrees are universal scaling limits, for the Gromov-Hausdorff topology, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter 3/2 is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.

math.PR↗
arXiv2014

The hull process of the Brownian plane

We study the random metric space called the Brownian plane, which is closely related to the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar triangulation. We obtain a number of explicit distributions for the Brownian plane. In particular, we consider, for every $r>0$, the hull of radius $r$, which is obtained by "filling in the holes" in the ball of radius $r$ centered at the root. We introduce a quantity $Z_r$ which is interpreted as the (generalized) length of the boundary of the hull of radius $r$. We identify the law of the process $(Z_r)_{r>0}$ as the time-reversal of a continuous-state branching process starting from $+\infty$ at time $-\infty$ and conditioned to hit $0$ at time $0$, and we give an explicit description of the process of hull volumes given the process $(Z_r)_{r>0}$. We obtain an explicit formula for the Laplace transform of the volume of the hull of radius $r$, and we also determine the conditional distribution of this volume given the length of the boundary. Our proofs involve certain new formulas for super-Brownian motion and the Brownian snake in dimension one, which are of independent interest.

math.PR↗
arXiv2014

A glimpse of the conformal structure of random planar maps

We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm--Loewner evolution) process of parameter $ κ= 6$ and to combine the locality property of the SLE_{6} together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension $\frac{1}{3}$ almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).

math.PR↗
arXiv2014

Scaling limits and influence of the seed graph in preferential attachment trees

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabási-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& Rácz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barabási-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension $2$.

math.PR↗
arXiv2014

Ergodic Theory on Stationary Random Graphs

A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic function. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.

math.PR↗
arXiv2014

The CRT is the scaling limit of random dissections

We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform $p$-angulations. As their number of vertices $n$ goes to infinity, we show that these random graphs, rescaled by $n^{-1/2}$, converge in the Gromov--Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees.

math.PR↗
arXiv2014

Planar stochastic hyperbolic infinite triangulations

Pursuing the approach of Angel & Ray, we introduce and study a family of random infinite triangulations of the full-plane that satisfy a natural spatial Markov property. These new random lattices naturally generalize Angel & Schramm's Uniform Infinite Planar Triangulation (UIPT) and are hyperbolic in flavor. We prove that they exhibit a sharp exponential volume growth, are non-Liouville, and that the simple random walk on them has positive speed almost surely. We conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size.

math.PR↗
arXiv2013

The local limit of unicellular maps in high genus

We show that the local limit of unicellular maps whose genus is proportional to the number of edges is a supercritical geometric Galton-Watson tree conditioned to survive. The proof relies on enumeration results obtained via the recent bijection given by the second author together with Feray and Fusy.

math.PR↗
arXiv2013

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 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.

math.PR↗
arXiv2012

The stable trees are nested

We show that we can construct simultaneously all the stable trees as a nested family. More precisely, if $1 < a < a' \leq 2$ we prove that hidden inside any a-stable we can find a version of an a'-stable tree rescaled by an independent Mittag-Leffler type distribution. This tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it. Our proofs are based on a recursive construction due to Marchal which is proved to converge almost surely towards a stable tree.

math.PR↗
arXiv2012

The Brownian plane

We introduce and study the random non-compact metric space called the Brownian plane, which is obtained as the scaling limit of the uniform infinite planar quadrangulation. Alternatively, the Brownian plane is identified as the Gromov-Hausdorff tangent cone in distribution of the Brownian map at its root vertex, and it also arises as the scaling limit of uniformly distributed (finite) planar quadrangulations with n faces when the scaling factor tends to 0 less fast than n^{-1/4}. We discuss various properties of the Brownian plane. In particular, we prove that the Brownian plane is homeomorphic to the plane, and we get detailed information about geodesic rays to infinity.

math.PR↗
arXiv2012

The Markovian hyperbolic triangulation

We construct and study the unique random tiling of the hyperbolic plane into ideal hyperbolic triangles (with the three corners located on the boundary) that is invariant (in law) with respect to Moebius transformations, and possesses a natural spatial Markov property that can be roughly described as the conditional independence of the two parts of the triangulation on the two sides of the edge of one of its triangles.

math.PR↗
arXiv2012

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 arXiv:1201.1052. 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 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 diffusive.

math.PR↗
arXiv2012

Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points

We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel to the quadrangulation case. Our main result is that, up to polylogarithmic factors, $n^3$ pioneer points have been discovered before the walk exits the ball of radius $n$ in the UIPQ. As a result we verify the KPZ relation in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1/3. Along the way, new geometric controls on the UIPQ are established.

math.PR↗
arXiv2012

Random recursive triangulations of the disk via fragmentation theory

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension $β^*+1$, where $β^*=(\sqrt{17}-3)/2$, and that it can be described as the geodesic lamination coded by a random continuous function which is Hölder continuous with exponent $β^*-\varepsilon$, for every $\varepsilon>0$. We also discuss recursive constructions of triangulations of the $n$-gon that give rise to the same continuous limit when $n$ tends to infinity.

math.PR↗
arXiv2012

Random non-crossing plane configurations: A conditioned Galton-Watson tree approach

We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution towards Aldous' Brownian triangulation of the disk. In the case of dissections, we also refine the study of the maximal vertex degree and validate a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of an underlying Galton-Watson tree structure.

math.PR↗