Meinardus' theorem on weighted partitions: Extensions and a probabilistic proof

Granovsky, Boris L.; Stark, Dudley; Erlihson, Michael

doi:10.1016/j.aam.2007.11.001

Cited by 29 publications

(85 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…using the saddle-point method [14], where s = σ + it is a complex variable. Granovsky, Stark, and Erlihson [17] extended Meinardus' theorem to new multiplicative combinatorial objects using Khintchine's probabilistic method [20]. Granovsky and Stark [16] later generalized their results to weighted partitions such that D(s) has multiple singularities on the positive real axis, which includes the class of Bose-Einstein condensates.…”

Section: Khintchine-meinardusprobabilisticmethodsmentioning

confidence: 99%

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

Bernstein¹,

Fahrbach²,

Randall³

2018

2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

View full text Add to dashboard Cite

Boltzmann sampling is commonly used to uniformly sample objects of a particular size from large combinatorial sets. For this technique to be effective, one needs to prove that (1) the sampling procedure is efficient and (2) objects of the desired size are generated with sufficiently high probability. We use this approach to give a provably efficient sampling algorithm for a class of weighted integer partitions related to Bose-Einstein condensation from statistical physics. Our sampling algorithm is a probabilistic interpretation of the ordinary generating function for these objects, derived from the symbolic method of analytic combinatorics. Using the Khintchine-Meinardus probabilistic method to bound the rejection rate of our Boltzmann sampler through singularity analysis of Dirichlet generating functions, we offer an alternative approach to analyze Boltzmann samplers for objects with multiplicative structure.

show abstract

Section: Khintchine-meinardusprobabilisticmethodsmentioning

confidence: 99%

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

Bernstein¹,

Fahrbach²,

Randall³

2018

2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

View full text Add to dashboard Cite

show abstract

“…There are more general Meinardus methods for computing asymptotics of the coefficients of classes of partition number generating functions of the form [10] n≥0…”

Section: 3mentioning

confidence: 99%

A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

Schmidt

2019

Axioms

View full text Add to dashboard Cite

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence's ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way.

show abstract

“…According to the representation (2.1) and independence of {D j } under the measure µ q z,k (see Section 2), the weight N(λ) of partition λ ∈ Λ q is the sum of k → ∞ independent random variables, so one may expect a local limit theorem to hold (cf. [6,41,16,17]). For our purposes, it suffices to obtain an asymptotic lower bound for the probability of the event {N(λ) = n}.…”

Section: A1 Auxiliary Lemmasmentioning

confidence: 99%

Limit Shape of Minimal Difference Partitions and Fractional Statistics

Bogachev

Yakubovich

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

The class of minimal difference partitions MDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q ≥ 0. In a recent series of papers by A. Comtet and collaborators, the MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classical Bose-Einstein (q = 0) and Fermi-Dirac (q = 1) cases. This was done by formally allowing values q ∈ (0, 1) using an analytic continuation of the limit shape of the corresponding Young diagrams calculated for integer q. To justify this "replica-trick", we introduce a more general model based on a variable MDP-type condition encoded by an integer sequence q = (q i ), whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of q. In this model, we find the family of limit shapes parameterized by q ∈ [0, ∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts.

show abstract

Meinardus' theorem on weighted partitions: Extensions and a probabilistic proof

Cited by 29 publications

References 18 publications

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

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

A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

Limit Shape of Minimal Difference Partitions and Fractional Statistics

Contact Info

Product

Resources

About