323 research outputs found
Tableaux and plane partitions of truncated shapes
We consider a new kind of straight and shifted plane partitions/Young
tableaux --- ones whose diagrams are no longer of partition shape, but rather
Young diagrams with boxes erased from their upper right ends. We find formulas
for the number of standard tableaux in certain cases, namely a shifted
staircase without the box in its upper right corner, i.e. truncated by a box, a
rectangle truncated by a staircase and a rectangle truncated by a square minus
a box. The proofs involve finding the generating function of the corresponding
plane partitions using interpretations and formulas for sums of restricted
Schur functions and their specializations. The number of standard tableaux is
then found as a certain limit of this function.Comment: Accepted to Advances in Applied Mathematics. Final versio
Hook formulas for skew shapes III. Multivariate and product formulas
We give new product formulas for the number of standard Young tableaux of
certain skew shapes and for the principal evaluation of the certain Schubert
polynomials. These are proved by utilizing symmetries for evaluations of
factorial Schur functions, extensively studied in the first two papers in the
series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We
also apply our technology to obtain determinantal and product formulas for the
partition function of certain weighted lozenge tilings, and give various
probabilistic and asymptotic applications.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook
formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317)
where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10
added, resembles published versio
Random sampling of plane partitions
This article presents uniform random generators of plane partitions according
to the size (the number of cubes in the 3D interpretation). Combining a
bijection of Pak with the method of Boltzmann sampling, we obtain random
samplers that are slightly superlinear: the complexity is in
approximate-size sampling and in exact-size sampling
(under a real-arithmetic computation model). To our knowledge, these are the
first polynomial-time samplers for plane partitions according to the size
(there exist polynomial-time samplers of another type, which draw plane
partitions that fit inside a fixed bounding box). The same principles yield
efficient samplers for -boxed plane partitions (plane partitions
with two dimensions bounded), and for skew plane partitions. The random
samplers allow us to perform simulations and observe limit shapes and frozen
boundaries, which have been analysed recently by Cerf and Kenyon for plane
partitions, and by Okounkov and Reshetikhin for skew plane partitions.Comment: 23 page
Free fermions in classical and quantum integrable models
This thesis studies the role of free fermions in certain classical and
quantum integrable models. The classical models studied are the KP and BKP
hierarchies of partial differential equations. We review the presence of free
fermions in the theory of these hierarchies, a topic which was established in
the work of the Kyoto school. The quantum models studied are descendents and
relatives of the XYZ model, or its lattice equivalent, the eight-vertex model.
We give a number of new results in the context of these models, revealing the
presence of fermions in typical quantities such as Bethe eigenvectors,
partition functions and scalar products.Comment: 2010 PhD thesis, Department of Mathematics and Statistics, University
of Melbourne. 293 page
Nonintersecting paths with a staircase initial condition
We consider an ensemble of discrete nonintersecting paths starting from
equidistant points and ending at consecutive integers. Our first result is an
explicit formula for the correlation kernel that allows us to analyze the
process as . In that limit we obtain a new general class of
kernels describing the local correlations close to the equidistant starting
points. As the distance between the starting points goes to infinity, the
correlation kernel converges to that of a single random walker. As the distance
to the starting line increases, however, the local correlations converge to the
sine kernel. Thus, this class interpolates between the sine kernel and an
ensemble of independent particles. We also compute the scaled simultaneous
limit, with both the distance between particles and the distance to the
starting line going to infinity, and obtain a process with number variance
saturation, previously studied by Johansson.Comment: 34 pages, 9 figures; reference added, Theorem 2.1 extended, typos
correcte
The Correlation Functions of the XXZ Heisenberg Chain for Zero or Infinite Anisotropy and Random Walks of Vicious Walkers
The XXZ Heisenberg chain is considered for two specific limits of the
anisotropy parameter: \Dl\to 0 and \Dl\to -\infty. The corresponding wave
functions are expressed by means of the symmetric Schur functions. Certain
expectation values and thermal correlation functions of the ferromagnetic
string operators are calculated over the base of N-particle Bethe states. The
thermal correlator of the ferromagnetic string is expressed through the
generating function of the lattice paths of random walks of vicious walkers. A
relationship between the expectation values obtained and the generating
functions of strict plane partitions in a box is discussed. Asymptotic estimate
of the thermal correlator of the ferromagnetic string is obtained in the limit
of zero temperature. It is shown that its amplitude is related to the number of
plane partitions.Comment: 22 pages, 1 figure, LaTe
Volume Laws for Boxed Plane Partitions and Area Laws for Ferrers Diagrams
We asymptotically analyse the volume-random variables of general, symmetric
and cyclically symmetric plane partitions fitting inside a box. We consider the
respective symmetry class equipped with the uniform distribution. We also prove
area limit laws for two ensembles of Ferrers diagrams. Most of the limit laws
are Gaussian.Comment: 9 pages, 1 figure, published in proceedings of the Fifth Colloquium
on Mathematics and Computer Scienc
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