Boltzmann Distribution on "Short" Integer Partitions with Power Parts: Limit Laws and Sampling

TL;DR

The paper analyzes two-parameter Boltzmann distributions on the space of strict partitions with parts in $q$-th powers, focusing on short-length regimes where the expected length $\langle M\rangle$ is fixed or grows slowly relative to the weight $\langle N\rangle$. It derives a suite of limit results, including Poisson and compound Poisson–Gamma limits for $(N_\lambda,M_\lambda)$, a central limit theorem, a gamma-based limit for the weight, and a universal limit shape for scaled Young diagrams $\omega_q^*(x)=1-G_{1/q}(x)$ in the slow-growth regime. The extremal-part analysis yields Weibull and Gumbel limits for the smallest and largest parts and confirms asymptotic independence across regimes; these results are complemented by exact and approximate cardinality formulas and subspace conditioning. The paper translates these probabilistic insights into practical sampling algorithms (free and rejection samplers), with bias-correction, truncation schemes, and complexity analyses, enabling efficient generation of random partitions in constrained spaces and enabling applications to testing representability problems and Gauss-circle-type counts. Overall, the work deepens the connection between combinatorial partition theory, limit theorems, and constructive sampling in a Boltzmann framework, with implications for statistical physics analogies and number-theoretic representations.

Abstract

The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set $\check{\varLambda}^{q}$ of strict integer partitions (i.e., with unequal parts) into perfect $q$-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters $\langle N\rangle$ and $\langle M\rangle$ controlling the expected weight and length, respectively. We study ``short'' partitions, where the parameter $\langle M\rangle$ is either fixed or grows slower than for typical plain (unconstrained) partitions. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed $\langle M\rangle$ and a limit shape result in the case of slow growth of $\langle M\rangle$. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyse their performance.

Figures (7)

• Figure 1: (a) The Young diagram $\varUpsilon_\lambda$ (shaded) of partition $\lambda = (10, 7, 5, 5, 4, 3, 1)$, with weight $N_\lambda = 35$ and length $M_\lambda = 7$. The graph of the step function $x\mapsto Y_\lambda(x)$ defined in \ref{['eq:Young']} depicts the upper boundary of $\varUpsilon_\lambda$ (shown in red in the online version). (b) Two classical limit shapes, for unrestricted partitions $\lambda\in\varLambda(n)$ (red) and strict partitions $\lambda\in\check{\varLambda}(n)$ (blue), determined by equations \ref{['eq:limit1']} and \ref{['eq:limit2']}, respectively.
• Figure 2: Geometric illustration of the sub-level partition sets $\check{\varLambda}^q_m(x)$ (defined in \ref{['eq:sub-level']}) with $m=2$ and (a) $q=1$ or (b) $q=2$, represented as the sets of integer points $(j_1,j_2) \in \mathbb{Z}^2$ such that $0<j_2<j_1$ and $j_1+j_2\le x$ or $j_1^2+j_2^2\le x$, respectively. In line with Theorem \ref{['th:ID']}, their cardinalities have the asymptotics $L_{2}(x)\sim \frac{1}{4}\hbox{$\:\!$} x^2$ and $L^2_{2}(x)\sim \frac{1}{8}\hbox{$\:\!$}\pi x$, corresponding to the area of the shaded domains.
• Figure 3: Illustration of convergence to the limit shape for $q=1$ and $q=2$ (in the online version shown in blue and red, respectively). The step plots depict the upper boundary of the scaled Young diagrams (see \ref{['eq:Y-tilde']}), while the smooth lines represent the limit shape $\omega_q^*(x)=1-G_{1/q}(x)$ (see \ref{['eq:omega']}). The corresponding partitions $\lambda\in\check{\varLambda}^q$ were sampled using Algorithm \ref{['sampler']} with hyper-parameters $\braket{M}=50$ and $\braket{N}=2.5\cdot 10^5$ ($q=1$) or $\braket{N}=1.25\cdot 10^7$ ($q=2)$; in both cases, $\kappa=0.01$ (cf. Assumption \ref{['as1']}). The respective sample weight and length are $N_\lambda=236{,}\hbox{$\:\!$}369$, $M_\lambda=52$ ($q=1$) and $N_\lambda=12{,}\hbox{$\:\!$} 733{,}\hbox{$\:\!$} 323$, $M_\lambda=45$ ($q=2$).
• Figure 4: Marginal histograms for the weight $N_\lambda$ (left) and length $M_\lambda$ (right) for random samples (of size $10^5$ each) from the partition space $\check{\varLambda}^q$ ($q=2$), simulated using a free Boltzmann sampler as set out in Algorithm \ref{['sampler']}. Color coding (online version): blue designates the limiting distributions under the "fixed" regime, that is, compound Poisson-Gamma (left) and Poisson (right); red indicates a normal approximation; magenta depicts a mean-gamma approximation (see Section \ref{['sec:4.1']}). In the black-and-white version, the normal curves on the left are identifiable by a noticeable positive shift.
• Figure 5: Joint sampling distribution of weight $N_\lambda$ and length $M_\lambda$ for $q=2$, $\braket{M}=50$ and $\braket{N}=10^7$ (cf. marginal plots in Figure \ref{['fig4']}(b)). A random Boltzmann sample of partitions $\lambda\in\check{\varLambda}^q$ (of size $10^5$) was simulated using Algorithm \ref{['sampler']}.

Theorems & Definitions (80)

• Definition 2.1
• Remark 2.1
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof