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Tessellations of random maps of arbitrary genus

Grégory Miermont (2009)

Annales scientifiques de l'École Normale Supérieure

We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these...

The chain records.

Gnedin, Alexander V. (2007)

Electronic Journal of Probability [electronic only]

The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

Ashkan Nikeghbali, Dirk Zeindler (2013)

Annales de l'I.H.P. Probabilités et statistiques

The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...

The logarithmic Sobolev constant of some finite Markov chains

Guan-Yu Chen, Wai-Wai Liu, Laurent Saloff-Coste (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

The logarithmic Sobolev constant is always bounded above by half the spectral gap. It is natural to ask when this inequality is an equality. We consider this question in the context of reversible Markov chains on small finite state spaces. In particular, we prove that equality holds for simple random walk on the five cycle and we discuss assorted families of chains on three and four points.

The "Thirty-seven Percent Rule" and the secretary problem with relative ranks

Béla Bajnok, Svetoslav Semov (2014)

Discussiones Mathematicae Probability and Statistics

We revisit the problem of selecting an item from n choices that appear before us in random sequential order so as to minimize the expected rank of the item selected. In particular, we examine the stopping rule where we reject the first k items and then select the first subsequent item that ranks lower than the l-th lowest-ranked item among the first k. We prove that the optimal rule has k ~ n/e, as in the classical secretary problem where our sole objective is to select the item of lowest rank;...

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