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Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation (Extended Abstract)
, 2012
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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.
Large unicellular maps in high genus
"... We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges. We prove that the distance between two uniformly selected vertices of such a map is of order log n and the diameter is also of ..."
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We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges. We prove that the distance between two uniformly selected vertices of such a map is of order log n and the diameter is also of order log n with high probability. We further prove a quantitative version of the result that the map is locally planar with high probability. The main ingredient of the proofs is an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy ([14]).
1 TAIL BOUNDS FOR THE HEIGHT AND WIDTH OF A RANDOM TREE WITH A GIVEN DEGREE SEQUENCE
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Asymptotics
, 2012
"... of trees with a prescribed degree sequence and applications ..."
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Finding paths in sparse random graphs requires many queries
"... We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph G ∼ G(n, p) in order to find a subgraph which possesses some target property with high probability. In this paper we focus o ..."
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We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph G ∼ G(n, p) in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in G ∼ G(n, p) when p = 1+εn for some fixed constant ε> 0. This random graph is known to have typically linearly long paths. To have ` edges with high probability in G ∼ G(n, p) one clearly needs to query at least Ω