Random permutations and their discrepancy process

Guillaume Chapuy (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

Mots clés: Weak convergence; Bivariate Brownian bridge; Random permutations

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Géométrie algorithmique [cs.CG]

Guillaume Chapuy. Random permutations and their discrepancy process. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. pp.457-470, ⟨10.46298/dmtcs.3534⟩. ⟨hal-01184782⟩