[Paper Review] Random sampling of plane partitions

Olivier Bodini, Éric Fusy+1 more|Dec 1, 2007

Bayesian Methods and Mixture Models3 citations

TL;DR

This paper presents the first polynomial-time uniform random samplers for plane partitions by size, using a bijection by Pak and Boltzmann sampling techniques. It achieves approximate-size sampling in O(n(ln n)^3) time and exact-size sampling in O(n^{4/3}) time, enabling efficient simulation of limit shapes and frozen boundaries in statistical physics models.

ABSTRACT

This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is $O(n (\ln n)^3)$ in approximate-size sampling and $O(n^{4/3})$ in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for $(a imes b)$-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.

Motivation & Objective

• To develop efficient uniform random samplers for plane partitions according to their size (total number of cubes in 3D interpretation).
• To overcome the lack of polynomial-time samplers for size-constrained plane partitions, unlike existing methods restricted to fixed bounding boxes.
• To enable large-scale simulations of statistical properties such as limit shapes and frozen boundaries observed in recent mathematical physics studies.
• To combine Pak’s bijection with Boltzmann sampling to achieve efficient sampling with good concentration properties.

Proposed method

• Use Pak’s bijection to map multisets of integer pairs (class M) to plane partitions, preserving size.
• Apply Boltzmann sampling to the class M, which has a simple combinatorial decomposition and generating function ∏_{r≥1} (1−x^r)^{-r}.
• Construct a Boltzmann sampler for M using symbolic combinatorics rules, enabling efficient random generation of multisets.
• Leverage the concentration of the size distribution under the Boltzmann model to design rejection-based approximate-size samplers.
• Use the inverse of Pak’s bijection to map sampled multisets to plane partitions.
• Apply complexity analysis based on expected running times, using asymptotic results for singular generating functions.

Experimental results

Research questions

• RQ1 Can uniform random sampling of plane partitions be achieved efficiently with respect to the size, rather than a fixed bounding box?
• RQ2 What is the computational complexity of exact-size and approximate-size uniform sampling for plane partitions?
• RQ3 How can the bijection by Pak be leveraged to reduce plane partition sampling to multiset sampling?
• RQ4 What are the asymptotic size concentration properties of the Boltzmann model for plane partitions?
• RQ5 Can the method be extended to (a×b)-boxed and skew plane partitions with similar efficiency?

Key findings

• The approximate-size sampler for plane partitions runs in O(n(ln n)^3) time under a real-arithmetic model.
• The exact-size sampler runs in O(n^{4/3}) time, marking the first polynomial-time exact-size sampler for plane partitions by size.
• The method achieves uniform sampling over all plane partitions of a given size, enabling statistical analysis of limit shapes and frozen boundaries.
• For (a×b)-boxed plane partitions, the expected running time is asymptotically ~ (ab / φ(ab)) n, where φ(α) = (α/e)^α / Γ(α).
• The size distribution under the Boltzmann model concentrates well, allowing efficient rejection sampling with high success probability.
• The framework extends naturally to (a×b)-boxed and skew plane partitions, with similar complexity bounds.