Pierre Lazag
In this note, we prove a law of large numbers for local patterns in plane partitions with geometric weight, a model introcued by Okounkov and Reshetikhin. Its proof is based on the determinantal structure of the corresponding point process, wich allows to control the decay of the correlations in a convenient manner.
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We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.
We give a simple proof of a nice formula for the means and covariances of the diagonal sums of a uniformly random boxed plane parition.
Plant differently colored points in the plane, then let random points ("Poisson rain") fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In the course of the proof, which heavily exploits time-reversals, we encounter a novel self-similar pr…
We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon's product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three two-dimensional families. For distributions from these families we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all …
We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP):…
Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well-known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in one-twelfth of a hexagon with free boundary to express them as perfect matchings of a family of non-bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit …
We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from (a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions. Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point pro…
Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in [OR2], but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in [OR2] is singular. We also obser…
In this paper we consider a probability distribution on plane partitions, which arises as a one-parameter generalization of the q^{volume} measure. This generalization is closely related to the classical multivariate Hall-Littlewood polynomials, and it was first introduced by Vuletic. We prove that as the plane partitions become large (q goes to 1, while the Hall-Littlewood parameter t is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy-Widom d…
For any partition of $\{1, 2, ..., n\}$ we define its {\it increments} $X_i, 1 \le i \le n$ by $X_i = 1$ if $i$ is the smallest element in the partition block that contains it, $X_i = 0$ otherwise. We prove that for partially exchangeable random partitions (where the probability of a partition depends only on its block sizes in order of appearance), the law of the increments uniquely determines the law of the partition. One consequence is that the Chinese Restaurant Process CRP($θ$) (the partition with distribution given by the Ewens sampling formula with parameter $θ$) is the only exchangeabl…
We study scaling limits of periodically weighted skew plane partitions with semilocal interactions and general boundary conditions. The semilocal interactions correspond to the Macdonald symmetric functions which are $(q,t)$-deformations of the Schur symmetric functions. We show that the height functions converge to a deterministic limit shape and that the global fluctuations are given by the $2$-dimensional Gaussian free field as $q,t\to 1$ and the mesh size goes to $0$. Specializing to the noninteracting case, this verifies the Kenyon-Okounkov conjecture for the case of the $r^{\mathrm{volum…
We study a single-flip dynamics for the monotone surface in (2+1) dimensions obtained from a boxed plane partition. The surface is analyzed as a system of non-intersecting simple paths. When the flips have a non-zero bias we prove that there is a positive spectral gap uniformly in the boundary conditions and in the size of the system. Under the same assumptions, for a system of size M, the mixing time is shown to be of order M up to logarithmic corrections.
We study the asymptotic distribution, as the volume parameter goes to 1, of the peak (largest part) of finite- or slowly-growing-width cylindric plane partitions weighted by their trace, seam, and volume. There are two natural asymptotic regimes depending on the trace/seam parameters, and in both cases we obtain asymptotics governed by finite temperature (periodic) analogues of the Bessel and Airy gap probabilities from random matrix theory. In particular, the distributions we obtain interpolate \emph{in more than one way} between two well-known extremal value distributions: the Gumbel distrib…
We study scaling limits of skew plane partitions with periodic weights under several boundary conditions. We compute the correlation kernel of the limiting point process in the bulk and near turning points on the frozen boundary. The turning points that appear in the homogeneous case split in our model into pairs of turning points macroscopically separated by a "semi-frozen" region. As a result the point process at a turning point is not the GUE minor process, but rather a pair of GUE minor processes, non-trivially correlated. We also study an intermediate regime when the weights are periodic …
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