398 research outputs found

    Perfect sampling algorithm for Schur processes

    Full text link
    We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entropy optimal (i.e. it reads a minimal number of random bits as an input). The algorithm encompasses previous growth procedures for special Schur processes related to the primal and dual RSK algorithm, as well as the famous domino shuffling algorithm for domino tilings of the Aztec diamond. It can be easily adapted to deal with symmetric Schur processes and general Schur processes involving infinitely many parameters. It is more concrete and easier to implement than Borodin's algorithm, and it is entropy optimal. At a technical level, it relies on unified bijective proofs of the different types of Cauchy and Littlewood identities for Schur functions, and on an adaptation of Fomin's growth diagram description of the RSK algorithm to that setting. Simulations performed with this algorithm suggest interesting limit shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints present in v2

    Monodromy and K-theory of Schubert curves via generalized jeu de taquin

    Get PDF
    We establish a combinatorial connection between the real geometry and the KK-theory of complex Schubert curves S(λ)S(\lambda_\bullet), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω\omega on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1\mathbb{RP}^1, with ω\omega as the monodromy operator. We provide a local algorithm for computing ω\omega without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the KK-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the KK-theory and real geometry of S(λ)S(\lambda_\bullet).Comment: 33 pages, 12 figures including 2 color figures; to appear in the Journal of Algebraic Combinatoric

    Lambda-determinants and domino-tilings

    Get PDF
    Consider the 2n2n-by-2n2n matrix M=(mi,j)i,j=12nM=(m_{i,j})_{i,j=1}^{2n} with mi,j=1m_{i,j} = 1 for i,ji,j satisfying 2i2n1+2j2n12n|2i-2n-1|+|2j-2n-1| \leq 2n and mi,j=0m_{i,j} = 0 for all other i,ji,j, consisting of a central diamond of 1's surrounded by 0's. When n4n \geq 4, the λ\lambda-determinant of the matrix MM (as introduced by Robbins and Rumsey) is not well-defined. However, if we replace the 0's by tt's, we get a matrix whose λ\lambda-determinant is well-defined and is a polynomial in λ\lambda and tt. The limit of this polynomial as t0t \to 0 is a polynomial in λ\lambda whose value at λ=1\lambda=1 is the number of domino tilings of a 2n2n-by-2n2n square.Comment: 4 pages; to appear in a special issue of Advances in Applied Mathematics honoring David P. Robbin

    Random Growth Models

    Full text link
    The link between a particular class of growth processes and random matrices was established in the now famous 1999 article of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. During the past ten years, this connection has been worked out in detail and led to an improved understanding of the large scale properties of one-dimensional growth models. The reader will find a commented list of references at the end. Our objective is to provide an introduction highlighting random matrices. From the outset it should be emphasized that this connection is fragile. Only certain aspects, and only for specific models, the growth process can be reexpressed in terms of partition functions also appearing in random matrix theory.Comment: Review paper; 24 pages, 4 figures; Minor correction

    Limits of Multilevel TASEP and similar processes

    Get PDF
    We study the asymptotic behavior of a class of stochastic dynamics on interlacing particle configurations (also known as Gelfand-Tsetlin patterns). Examples of such dynamics include, in particular, a multi-layer extension of TASEP and particle dynamics related to the shuffling algorithm for domino tilings of the Aztec diamond. We prove that the process of reflected interlacing Brownian motions introduced by Warren in \cite{W} serves as a universal scaling limit for such dynamics.Comment: 16 pages, 1 figur
    corecore