Random permutations and their discrepancy process

نویسنده

  • Guillaume Chapuy
چکیده

On the one hand, random permutations appear as a natural model for data in various topics, such as analysis of algorithms, statistical mechanics, or genomic statistics. On the other hand, from the mathematical point of view, various parameters of random permutations have been studied in combinatorics and probability, like the distribution of the lengths of cycles, the repartition of the eigenvalues on the unit circle([Wie00]), or the length of large monotonic subsequences (see [BDJ99, AD99]). The description of the typical behaviour of these statistics provide powerful tools for applications, like the construction of statistical tests, or precise analysis of the execution time of algorithms. A lot of examples of permutations statistics and their application to analysis of algorithms can be found all along the book [FS]. The purpose of this paper is to describe a new ”process-valued” statistic of random permutations, recently introduced in the context of computational biology ([YBSS06]) to extract information from the large amounts of data produced by the microarray technology. Roughly speaking, a DNA microarray provides, for a given list of genes (say 1, . . . n), an ordering (i.e., a permutation) τ1, . . . , τn, of the genes by their expression level in a given experimental condition. Here, the typical value of n is 1000. In [YBSS06], the following problem is addressed: being given two such orderings τ and τ ′ corresponding to two classes of experiences (say on healthy and sick patients), is it possible to quantify the similarity

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تاریخ انتشار 2008