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Cycle structure of percolation on high-dimensional tori

Remco van der Hofstad, Artëm Sapozhnikov (2014)

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

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdős–Rényi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structureof the largest...

Determinantal probability measures

Russell Lyons (2003)

Publications Mathématiques de l'IHÉS

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation,...

Digital search trees and chaos game representation*

Peggy Cénac, Brigitte Chauvin, Stéphane Ginouillac, Nicolas Pouyanne (2009)

ESAIM: Probability and Statistics

In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which one can visualize repetitions of subwords (seen as suffixes of subsequences). The CGR-tree turns a sequence of letters into a Digital Search Tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height, the insertion depth for DST built from independent successive random sequences having the same distribution. Here the successive inserted words...

Discrete limit laws for additive functions on the symmetric group

Eugenijus Manstavičius (2005)

Acta Mathematica Universitatis Ostraviensis

Inspired by probabilistic number theory, we establish necessary and sufficient conditions under which the numbers of cycles with lengths in arbitrary sets posses an asymptotic limit law. The approach can be extended to deal with the counts of components with the size constraints for other random combinatorial structures.

Entropy of Schur–Weyl measures

Sevak Mkrtchyan (2014)

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

Relative dimensions of isotypic components of N th order tensor representations of the symmetric group on n letters give a Plancherel-type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when N n converges to a constant. The main...

Exact distribution under independence of the diagonal section of the empirical copula

Arturo Erdely, José M. González–Barrios (2008)

Kybernetika

In this paper we analyze some properties of the empirical diagonal and we obtain its exact distribution under independence for the two and three- dimensional cases, but the ideas proposed in this paper can be carried out to higher dimensions. The results obtained are useful in designing a nonparametric test for independence, and therefore giving solution to an open problem proposed by Alsina, Frank and Schweizer [2].

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