2,057 research outputs found
From Aztec diamonds to pyramids: steep tilings
We introduce a family of domino tilings that includes tilings of the Aztec
diamond and pyramid partitions as special cases. These tilings live in a strip
of of the form for some integer , and are parametrized by a binary word that
encodes some periodicity conditions at infinity. Aztec diamond and pyramid
partitions correspond respectively to and to the limit case
. For each word and for different types of boundary
conditions, we obtain a nice product formula for the generating function of the
associated tilings with respect to the number of flips, that admits a natural
multivariate generalization. The main tools are a bijective correspondence with
sequences of interlaced partitions and the vertex operator formalism (which we
slightly extend in order to handle Littlewood-type identities). In
probabilistic terms our tilings map to Schur processes of different types
(standard, Pfaffian and periodic). We also introduce a more general model that
interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6,
new improved proof of Proposition 11
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
Partial transpose of random quantum states: exact formulas and meanders
We investigate the asymptotic behavior of the empirical eigenvalues
distribution of the partial transpose of a random quantum state. The limiting
distribution was previously investigated via Wishart random matrices indirectly
(by approximating the matrix of trace 1 by the Wishart matrix of random trace)
and shown to be the semicircular distribution or the free difference of two
free Poisson distributions, depending on how dimensions of the concerned spaces
grow. Our use of Wishart matrices gives exact combinatorial formulas for the
moments of the partial transpose of the random state. We find three natural
asymptotic regimes in terms of geodesics on the permutation groups. Two of them
correspond to the above two cases; the third one turns out to be a new matrix
model for the meander polynomials. Moreover, we prove the convergence to the
semicircular distribution together with its extreme eigenvalues under weaker
assumptions, and show large deviation bound for the latter.Comment: v2: change of title, change of some methods of proof
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
General Fragmentation Trees
We show that the genealogy of any self-similar fragmentation process can be
encoded in a compact measured real tree. Under some Malthusian hypotheses, we
compute the fractal Hausdorff dimension of this tree through the use of a
natural measure on the set of its leaves. This generalizes previous work of
Haas and Miermont which was restricted to conservative fragmentation processes
- …