In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab., vol. 39, pp. 2224‐2270, 2011] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process M . Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov‐Hausdorff sense to a limit real tree T , which is encoded by M . This confirms a conjecture of Curien and Le Gall. 1 Introduction and main results In [18], Curien and Le Gall introduce the model of random recursive triangulations of the disk. The construction goes as follows: Atn = 1, two points are sampled independently with uniform distribution on the circle. They are connected by a chord (a straight line) which splits the disk into two fragments. Later on, at each step, two independent points are sampled uniformly at random on the circle and are connected by a chord if the latter does not intersect any of the previously inserted chords; in other words the two points are connected by a chord if they both fall in the same fragment. This gives rise to a sequence of laminations of the disk; for us a lamination will be a collection of chords which may only intersect at their end points. At timen the lamination Ln consists of the union of the chords inserted up to timen. As an increasing closed subset of the disk, Ln converges, and it is proved in [18] that
The work of was motivated by the pioneering work of who studied of the disk which arise as limiting objects for uniform triangulations of regular @math -gons as @math . In the case of uniform random triangulations, the process which encodes the limit triangulation is the Brownian excursion, and the scaling limit of the sequence of dual trees is the Brownian continuum random tree introduced in @cite_7 @cite_23 @cite_31 .
@cite_31 @cite_23 @cite_7
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Consider a family of random ordered graph trees (T-n)(n >= 1), where T-n has n vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as n -> infinity, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be resealed to converge to the Brownian motion on a related alpha-stable tree. We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses. We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web. We identify the scaling limit for the sizes of the largest components at criticality for inhomogeneous random graphs with weights that have finite third moments. We show that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, which extends results of Aldous (1997) for the critical behavior of Erdos-Renyi random graphs. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme initiated in van der Hofstad (2009) to study the near-critical behavior in inhomogeneous random graphs of so-called rank-1.