Size distribution of the largest component of a random A-mapping

Abstract

Abstract Let S n $ \mathfrak S_n $ be a semigroup of all mappings from the n-element set X into itself. We consider a set S n ( A ) $ \mathfrak S_n(A) $ of mappings from S n $ \mathfrak S_n $ such that their contour sizes belong to the set A ⊆ N. These mappings are called A-mappings. Let a random mapping τ n have a distribution on S n ( A ) $ \mathfrak S_n(A) $ such that each connected component with volume i ∈ N have weight ϑ i ⩾0. Let D be a subset of N. It is assumed that ϑ i  → ϑ>0 for i ∈ D and ϑ i  → 0 for i ∈ N∖ D as i → ∞. Let μ(n) be the maximal volume of components of the random mapping τ n . We suppose that sets A and D have asymptotic densities ϱ>0 and ρ>0 in N respectively. It is shown that the random variables μ(n)/n converge weakly to random variable ν as n → ∞. The distribution of ν coincides with the limit distribution of the corresponding characteristic in the Ewens sampling formula for random permutation with the parameter ρ ϱ ϑ/2.