Combinatorics: Ofir Gorodetsky (TAU)

Mon, 18/11/201910:00-12:00
C-400, CS building

Speaker: Ofir Gorodetsky (TAU)

Title: The Anatomy of Integers and Ewens Permutations

We will discuss an analogy between integers and permutations, an analogy which goes back to works of Erdős and Kac and of Billingsley which we shall survey. Certain statistics of the prime factors of a uniformly drawn integer (between 1 and x) agree, in the limit, with similar statistics of the cycles of a uniformly drawn permutation from the symmetric group on n elements. This analogy is beneficial to both number theory and probability theory, as one can often prove new number-theoretical results by employing probabilistic ideas, and vice versa.

The Ewens measure with parameter θ, first discovered in the context of population genetics, is a non-uniform measure on permutations. We will present an analogue of this measure on the integers, and show how natural questions on the integers have answers which agree with analogous problems for the Ewens measure. For example, the size of the prime factors of integers which are sums of two squares, and the cycle lengths of permutations drawn according to the Ewens measure with parameter 1/2, both converge to the Poisson-Dirichlet process with parameter 1/2. We will convey some of the ideas behind the proofs.

Joint work with Dor Elboim.