Ergodicity of multiplicative statistics

Abstract

For a subfamily of multiplicative measures on integer partitions we give conditions for properly rescaled associated Young diagrams to converge in probability to a certain deterministic curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.

Asymptotic enumeration by Khintchine–Meinardus probabilistic method: necessary and sufficient conditions for sub-exponential growth

“…with a given integer l ≥ 1 and with a function L analytic in the unit disk and such that 0 < |L(z)| < ∞, |z| ≤ 1. The assumption (21) conforms to the one by Yakubovich in [24], in the particular case z 0 = 1 and L is a slowly varying function. Assumption (21) extends our study to models with more general S(z), e.g.…”

“…Remark In connection with (59) it is in order to note that in the theory of limit shapes the parameter δ (called there scaling) is taken to be equal O(n − 1 ρr +1 ), which is, roughly speaking, (59) (see e.g [4], [23], [24]). The aforementioned coincidence is explained by the fact that the derivation of limit shapes consists of asymptotic approximation of probabilities with respect to the same multiplicative measure µ n as in our setting.…”

Improvements to exact Boltzmann sampling using probabilistic divide-and-conquer and the recursive method

“…There are many quantitative reasons why accepting a random sample of a random size serves as a good surrogate for an exact sample. A primary example is the limit shape of integer partitions, see [27], where it was shown that the limit shape of integer partitions coincides with the limit shape obtained by a Boltzmann model; see also [7,9,20,35] for related results.…”

Analyzing Boltzmann Samplers for Bose–Einstein Condensates with Dirichlet Generating Functions

“…This algorithm generates samples of size n exactly from the uniform distribution in expected polynomial time. This allows us, for example, to rigorously study the expected width of Young diagrams arising from random configurations (or, equivalently, the fraction of particles in a BEC in their ground state) without relying on the limiting properties given in [25].…”