Equivalence of ensembles for random plane partitions: passing from the $q^{\mathrm{volume}}$ measure to the uniform measure at fixed volume

Okounkov and Reshetikhin (2003) described the local geometry of a large random 3-dimensional Young diagram (a plane partition stacked in the positive octant) under the measure $M_q$ that weights a plane partition $\pi$ by $q^{|\pi|}$: the diagonal slices form a determinantal point process, and as $q\to 1$ the local statistics converge to the incomplete beta kernel in the disordered region (degenerating to the discrete sine kernel on diagonal slices), with Airy kernels at the facet boundaries established in later work. Section 3.2.6 of that paper states that passing from these $q\to 1$ asymptotics to the asymptotics of the uniform measure on plane partitions of a fixed large volume $N$ — the equivalence of ensembles — needs further work. The volume constraint couples all diagonal slices, so the fixed-volume (microcanonical) ensemble is not itself a Schur process and the determinantal machinery does not directly apply. The ensemble-equivalence transfer itself — the same limit shape and the same local determinantal statistics under the uniform fixed-volume measure as under $M_q$ at the matching volume scale — remains open.

Background

A plane partition is a two-dimensional array $\pi(i,j)$ of non-negative integers that is weakly decreasing along each row and each column; equivalently, it is a stack of unit cubes in the positive octant of $\mathbb{R}^3$ piled against the coordinate planes, called a solid or 3-dimensional Young diagram. The volume $|\pi|$ is the total number of cubes. The interface of such a stack projects to a lozenge tiling (a tiling by unit rhombi with angles $60^\circ$ and $120^\circ$) of the triangular lattice, which is the standard setting for random-surface models.

A natural probability measure on all plane partitions is the grand-canonical (or $q^{\mathrm{volume}}$) measure

$M_q(\pi)=\frac{q^{|\pi|}}{Z(q)},\qquad Z(q)=\prod_{k\ge 1}(1-q^k)^{-k},$

where the product formula for $Z(q)$ is MacMahon's classical enumeration of plane partitions. As $q\to 1$, typical diagrams have volume of order $(1-q)^{-3}$ and, after rescaling, converge to a deterministic limit shape with two kinds of regions: frozen (faceted) regions where the tiling is periodic, and a disordered (liquid) region where all lozenge types mix.

Okounkov and Reshetikhin (2003) analyzed this ensemble through the Schur process, a time-dependent version of the Schur measure. (A Schur function $s_\lambda$ is a symmetric polynomial indexed by a partition $\lambda$, the character of the corresponding irreducible representation of the symmetric groups; the Schur measure weights a partition $\lambda$ proportionally to $s_\lambda(X)s_\lambda(Y)$ for two sets of variables. The Schur process extends this to a measure on whole sequences of partitions.) They proved that the diagonal slices of a random 3-dimensional Young diagram under $M_q$ form a determinantal point process — a random point process whose correlation functions (joint intensities) are determinants $\det[K(x_i,x_j)]$ of an explicit correlation kernel given by a double contour integral — and analyzed its $q\to 1$ asymptotics. The limiting kernel on the two-dimensional lattice of the diagram is expressed through the incomplete beta function and generalizes the discrete sine kernel $\sin(\theta(x-y))/(\pi(x-y))$, which is what appears on a single diagonal slice; in the frozen regions the kernel degenerates to a Kronecker delta. Ferrari and Spohn (2003), using the techniques of that paper, showed that at the boundary between a frozen facet and the disordered region the edge scaling limit is governed by the Airy kernel and the Airy process (the universal edge process of random matrix theory built from the Airy function $\mathrm{Ai}$, the solution of $y''=xy$ with oscillatory decay); Okounkov and Reshetikhin (2006) later established the sine kernel in the bulk, the Airy kernel near a general point of the frozen boundary, and a Pearcey kernel at cusps for skew 3-dimensional partitions with the same $q^{\mathrm{volume}}$ weight.

All of these results concern the $q$-weighted, grand-canonical ensemble. The equally natural microcanonical ensemble — the uniform measure $U_N$ on plane partitions of a fixed volume $N$ — is not covered. In statistical mechanics, the equivalence of ensembles is the principle that conditioning a grand-canonical ensemble on the value of a conserved quantity yields, in the thermodynamic limit, the same behavior as the microcanonical ensemble with that value of the constraint. For the Plancherel measure on ordinary (one-dimensional) partitions, the analogous transfer was carried out by Borodin, Okounkov, and Olshanski (2000) as a depoissonization step (removing a Poisson averaging under which the measure becomes determinantal). Okounkov and Reshetikhin (2003, Section 3.2.6) note this precedent and state that in the plane-partition setting 'further work is needed to verify the equivalence of ensembles in the asymptotics'. The difficulty is structural: the total volume is a global constraint that couples all diagonal slices simultaneously, so $U_N$ is not itself a Schur process and has no known determinantal contour-integral representation. For comparison, Cohn, Larsen, and Propp (1998) determined the limit shape of uniform plane partitions in a box, where the box boundary plays the role of the constraint and the slices decouple in a different way; the free-octant fixed-volume case has no such decoupling. The question is whether the limit shape and the local determinantal statistics established for $M_q$ as $q\to 1$ are shared by $U_N$ as $N\to\infty$.

Problem Statement

Let $U_N$ denote the uniform probability measure on plane partitions $\pi$ (solid Young diagrams in the positive octant) of fixed volume $|\pi|=N$, and let $M_q$ denote the grand-canonical measure proportional to $q^{|\pi|}$ over all plane partitions. Okounkov and Reshetikhin (2003, Section 3.2.6) pose the equivalence of ensembles for these two ensembles: prove, or disprove, that the $N\to\infty$ asymptotics of $U_N$ coincide with the $q\to 1$ asymptotics of $M_q$ at the corresponding volume scale. The equivalence consists of three conditions:

1. A coupling of the parameters, for example $q=q(N)$ chosen so that the $M_q$-typical volume is of order $N$, under which the $N\to\infty$ asymptotics of $U_N$ are insensitive to the choice of coupling within the natural class, or under which the couplings that are covered are specified precisely.
2. The macroscopic limit: after the same rescaling as in the $M_q$ analysis, a plane partition distributed according to $U_N$ converges in probability to the same deterministic limit shape, with the same frozen and disordered regions.
3. The microscopic limit: the local statistics of $U_N$ around points of the rescaled limit shape converge to the same limiting kernels as for $M_q$ — the incomplete beta kernel in the disordered region (including its degeneration to the discrete sine kernel on diagonal slices) and the Airy kernel at the facet boundary.

The central case is the free (unboxed) octant geometry studied by Okounkov and Reshetikhin. A disproof is an admissible answer: it would have to exhibit, with complete proofs of the asymptotics on both sides, a genuine discrepancy between the two ensembles — for example a different limiting local kernel at a specified location and scaling, or a different limit shape — that is not an artifact of inconsistent normalizations or mismatched couplings.

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