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
Among the recent work on laminations of the disk, one can mention @cite_27 where Curien and Kortchemski showed that the Brownian triangulation is also the scaling limit of other random subsets of the disk, in particular non-crossing trees (sets of non-crossing chords which form a tree) @cite_8 , and dissections (non-crossing sets of chords) under the uniform distribution. By sampling tessellations according to a Boltzmann weight depending on the degree of the faces, obtained limit laminations which are not triangulations and are encoded by excursions of stable spectrally positive L 'evy processes (with L 'evy measure concentrated on @math ). Finally, have studied geodesic laminations of the Poincar 'e disk. They construct and study the unique random tiling of the hyperbolic plane into triangles with vertices on the boundary whose distribution is invariant under M "obius transformations and satisfies a certain spatial Markov property.
@cite_27 @cite_8
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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. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 236-260, 2014 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 Mobius 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. The following question is due to Chatterjee and Varadhan 2011. Fix 0<p<r<1 and take G Gn,p, the Erdi¾?s-Renyi random graph with edge density p, conditioned to have at least as many triangles as the typical Gn,r. Is G close in cut-distance to a typical Gn,r? Via a beautiful new framework for large deviation principles in Gn,p, Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of p,r where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d-regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all p,r such that rd,hpr lies on the convex minorant of xi¾?hpx1 d where hp is the rate function of a binomial with parameter p. In particular, the answer for triangles involves hpx rather than the natural guess of hpx1 3 where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of G Gn,p is conditioned to exceed the typical value of the largest eigenvalue of Gn,r. Building on the work of Chatterjee and Diaconis 2012 we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn 2001 and Galvin and Tet ali 2004. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109-146, 2015 This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers.