Introduction

In last few years there appeared a number of papers indicating a strong connection of certain asymptotic problems of enumerative combinatorics and representation theory of symmetric groups with the random matrix theory; see [Baik et al. 1999a; 1999b; Baik and Rains 1999a; 1999b; Borodin 1998a; 1998b; 1999; ≥ 2001; Borodin and Olshanski 1998a; 1998b; 2000a; Borodin et al. 2000; Johansson 2000; 1999; Okounkov 1999b; 1999a; Olshanski 1998a; 1998b; Tracy and Widom 1998; 1999], for a partial list. Such a connection was also anticipated in earlier works [Regev 1981; Kerov 1993; 1994]. For other interesting connections see also [Borodin 2000b; Borodin and Okounkov 2000; Okounkov 2001].

In this paper we suggest a hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with so-called “β = 2 ensembles” arising in random matrix theory. (These ensembles are characterized by the property that their correlation functions have determinantal form with a scalar kernel; see below.) We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group; see [Kerov et al. 1993] and Section 3 below.