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Percolation on random triangulations and stable looptrees
 In preparation
"... We study site percolation on Angel & Schramm’s Uniform Infinite Planar Triangulation. We compute several critical and nearcritical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in partic ..."
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Cited by 9 (4 self)
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We study site percolation on Angel & Schramm’s Uniform Infinite Planar Triangulation. We compute several critical and nearcritical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3/2, which was introduced in [13]. We also give a conjecture linking looptrees of any index α ∈ (1, 2) with scaling limits of cluster boundaries in random triangulations decorated with O(N) models. Figure 1: A site percolated triangulation and the interfaces separating the clusters.
Rescaled bipartite planar maps converge to the Brownian map
 In preparation
, 2013
"... Abstract. For every integer n ≥ 1, we consider a random planar map Mn which is uniformly distributed over the class of all rooted bipartite planar maps with n edges. We prove that the vertex set ofMn equipped with the graph distance rescaled by the factor (2n)−1/4 converges in distribution, in the G ..."
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Cited by 6 (0 self)
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Abstract. For every integer n ≥ 1, we consider a random planar map Mn which is uniformly distributed over the class of all rooted bipartite planar maps with n edges. We prove that the vertex set ofMn equipped with the graph distance rescaled by the factor (2n)−1/4 converges in distribution, in the GromovHausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map. 1.
The scaling limit of random outerplanar maps
, 2014
"... A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with n vertices suitably rescaled by a factor ..."
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Cited by 2 (0 self)
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A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with n vertices suitably rescaled by a factor
Scaling limits of random graphs from subcritical classes
, 2014
"... We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cn/ n converges to the Brownian Continuum Random Tree Te multiplied by a constant scaling factor that depends on the class under consideration. In add ..."
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We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cn/ n converges to the Brownian Continuum Random Tree Te multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter D(Cn) and height H(C n) of the rooted random graph C n. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on Cn, where we show the convergence to Te under an appropriate rescaling.
figure shows a large critical Galton–Watson tree with finite variance.
, 2013
"... We study a particular type of subcritical Galton–Watson trees, which are called nongeneric trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned nongeneric trees, meaning that with high probability th ..."
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We study a particular type of subcritical Galton–Watson trees, which are called nongeneric trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned nongeneric trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phenomenon by studying scaling limits of such trees and show that the situation is completely different from the critical case. In particular, the height of such trees grows logarithmically in their size. We also study fluctuations around the condensation vertex.
Trees and spatial topology change in CDT
"... Generalized causal dynamical triangulations (generalized CDT) is a model of twodimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. We solve the model at the discretized level using bijections between quadrangulations and trees. In the continuum lim ..."
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Generalized causal dynamical triangulations (generalized CDT) is a model of twodimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. We solve the model at the discretized level using bijections between quadrangulations and trees. In the continuum limit (scaling limit) the amplitudes are shown to agree with known formulas and explicit expressions are obtained for loop propagators and twopoint functions. It is shown that from a combinatorial point of view generalized CDT can be viewed as the scaling limit of planar maps with a finite number of faces and we determine the distance function on this ensemble of planar maps. Finally, the relation with planar maps is used to illuminate a mysterious identity of certain continuum cylinder amplitudes.