398 research outputs found
Perfect sampling algorithm for Schur processes
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
We establish a combinatorial connection between the real geometry and the
-theory of complex Schubert curves , 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 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 , with as the monodromy operator.
We provide a local algorithm for computing 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
-theoretic Littlewood-Richardson coefficient associated to the Schubert
curve. We then give purely combinatorial proofs of several numerical results
involving the -theory and real geometry of .Comment: 33 pages, 12 figures including 2 color figures; to appear in the
Journal of Algebraic Combinatoric
Lambda-determinants and domino-tilings
Consider the -by- matrix with for satisfying and for all
other , consisting of a central diamond of 1's surrounded by 0's. When , the -determinant of the matrix (as introduced by Robbins
and Rumsey) is not well-defined. However, if we replace the 0's by 's, we
get a matrix whose -determinant is well-defined and is a polynomial in
and . The limit of this polynomial as is a polynomial in
whose value at is the number of domino tilings of a
-by- square.Comment: 4 pages; to appear in a special issue of Advances in Applied
Mathematics honoring David P. Robbin
Random Growth Models
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
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
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