[Paper Review] The shape of a typical boxed plane partition

Henry Cohn, Michael Larsen+1 more|ArXiv.org|Jan 13, 1998

graph theory and CDMA systemsEngineering13 references215 citations

TL;DR

This paper determines the limiting shape of a typical boxed plane partition in a large rectangular box using a calculus of variations approach, showing that the solid Young diagram converges to a deterministic surface governed by the ratios of the box dimensions. The analysis connects plane partitions to lozenge tilings of hexagons and derives the typical height function via variational optimization of a functional derived from MacMahon's product formula generalization.

ABSTRACT

Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux.

Motivation & Objective

• To determine the asymptotic shape of a uniformly random plane partition constrained within a large a×b×c box.
• To establish a connection between plane partitions and lozenge tilings of equiangular hexagons, enabling geometric and probabilistic analysis.
• To generalize MacMahon’s formula for counting plane partitions to include constraints on tiling behavior along specific lines.
• To formulate a variational problem that characterizes the typical height function of the solid Young diagram in the limit of large box sizes.
• To analyze the resulting functional using methods adapted from Logan–Shepp and Vershik–Kerov on random Young tableaux.

Proposed method

• Use a bijection between boxed plane partitions and lozenge tilings of an a,b,c hexagon to translate combinatorial problems into tiling geometry.
• Generalize MacMahon’s product formula to count tilings with prescribed behavior on a fixed horizontal line, enabling conditional probability analysis.
• Construct a functional derived from the logarithm of the generalized product formula, which is then approximated by a Riemann sum.
• Convert the discrete sum into a continuous double integral using a carefully constructed function C to control the inverse derivative behavior and ensure convergence.
• Apply calculus of variations to maximize the limiting functional, yielding the typical height function as the solution to a variational problem.
• Use techniques from random matrix theory and asymptotic analysis, particularly those of Logan–Shepp and Vershik–Kerov, to solve the variational problem and derive the limit shape.

Experimental results

Research questions

• RQ1 What is the limiting shape of a typical plane partition in a large a×b×c box, and how does it depend on the ratios of a, b, and c?
• RQ2 How can the distribution of lozenge orientations in a random tiling of a large hexagon be characterized asymptotically?
• RQ3 What is the functional form of the height function that maximizes the asymptotic logarithmic weight of plane partitions in a box?
• RQ4 How does the generalized MacMahon formula for restricted tiling behavior relate to the emergence of a deterministic limit shape?
• RQ5 What conditions ensure that the Riemann sum approximation of the log-product formula converges to a well-defined integral in the limit?

Key findings

• The typical shape of a boxed plane partition converges to a deterministic surface in the large-box limit, with the shape depending only on the ratios of the box dimensions a:b:c.
• The limiting height function is derived as the solution to a variational problem involving a double integral of log differences of inverse cumulative distribution functions.
• The difference between the discrete sum and its continuous integral approximation is o(1), ensuring the validity of the asymptotic analysis.
• The functional to be maximized is constructed from a generalized product formula that counts plane partitions with specified behavior along a line, extending MacMahon’s classical result.
• The analysis confirms that the typical tiling exhibits a well-defined, smooth limit shape with three distinct lozenge orientations distributed according to a deterministic density.
• The method successfully adapts techniques from random Young tableaux to the setting of plane partitions, yielding a rigorous derivation of the limit shape.