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

Abstract

In this paper we prove the necessity of the main sufficient condition of Meinardus for sub exponential rate of growth of the number of structures, having multiplicative generating functions of a general form and establish a new necessary and sufficient condition for normal local limit theorem for aforementioned structures. The latter result allows to encompass in our study structures with sequences of weights having gaps in their support.

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

“…Probabilistic interpretations of ordinary generating functions for weighted partitions are useful for producing samples from Boltzmann distributions in polynomial time [2,13], but rejection sampling is not always guaranteed to be efficient. We use the Khintchine-Meinardus probabilistic method [15,16,17] to establish that rejection sampling is efficient for a broad class of weighted partitions that includes BECs. We also use the Boltzmann sampling framework based on the symbolic method from analytic combinatorics to give a improved algorithms for BECs, instead of simply using a geometric random variable for each degenerate energy state (Theorem 1.1 vs. Theorem 1.2 with r = 2).…”

Explicit asymptotic formulae for sub exponentially growing multiplicative structures

“…Our starting point is the following mathematical formalism adopted from [2] and from the preceding works [4], [5] with Dudley Stark.…”

“…For more details and references see the recent paper [9]. In [2] there were formulated necessary and sufficient conditions providing sub exponential growth of c n , n → ∞, given by the following asymptotic formula:…”

“…n √ 2π comes from the normal local limit theorem which is a part of Khintchine's representation of c n (see the aforementioned references). It was proven in [2] that for local limit theorem to hold it is sufficient that the weights {b j } obey the condition 1≤k≤n,q |k b k ≥ O(log 2 n), for any integer q > 1.…”

“…The purpose of the present paper is to derive the explicit expression in n of the RHS of the asymptotic formula (2). In [4] the asymptotic formula for c n was expressed via powers of n with coefficients defined recursively, i.e.…”