Random partitions with restricted part sizes

Abstract

ABSTRACT:For a subset S of positive integers let (n, S) be the set of partitions of n into summands that are elements of S. For every λ ∈ (n, S), let M n (λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on (n, S), and regard M n as a random variable. In this paper the limiting density of the (suitably normalized) random variable M n is determined for sets that are sufficiently regular. In particular, our results cover the case S = {Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman's coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes.

Ergodicity of multiplicative statistics

“…The distribution of the number of parts in such partitions has been studied recently in [11] under some assumptions on growth of B k . Namely its is shown that if B k − ck β , β ∈ (0, 1), satisfies some additional condition then the number of parts in a random partition of n behaves like a nondegenerate random variable (explicitly specified in [11]) multiplied by n 1/(1+β) .…”

“…The distribution of the number of parts in such partitions has been studied recently in [11] under some assumptions on growth of B k . Namely its is shown that if B k − ck β , β ∈ (0, 1), satisfies some additional condition then the number of parts in a random partition of n behaves like a nondegenerate random variable (explicitly specified in [11]) multiplied by n 1/(1+β) . Theorem 6 shows that if just summands greater than tn 1/(β+1) , t > 0 are counted then their number is much less: it is proportional to n β/(1+β) and the coefficient converges in probability to a constant (depending on t).…”

“…(The probabilist would say that a random partition of n is just a sequence of independent random variables with specific distributions conditioned to have some weighted sum equal n.) This technique seems to be first applied to random partitions by B. Fristedt [10]; now it is often referred to as Fristedt's conditional device. Later it has been frequently used by various authors in the related problems, see [14,6,11], to name just a few references. Vershik [20] noted that the similar technique is applicable to a wider range of problems with the same property that the measure is a product measure restricted to a certain affine subset.…”

Limit Shapes via Bijections

“…The weak form of the limit shape in Theorem 3.3 is easily calculated via the statistical models in Khinchin [35], and more explicitly by Kerov and Vershik [56] -the same ones utilized by Fristedt [24] and Pittel [45]. The limit shape then follows by concentration and a local central limit theorem, or equivalently, asymptotic enumeration, via the inequality in equation (10.6); see also [27,12].…”

On Tails of Perpetuities

“…Specifically, we prove Proposition 1. Assume |Q| ≤ q, |M| ≤ 1, and let R be given by (1). Then, for sufficiently large x…”