2,057 research outputs found

    From Aztec diamonds to pyramids: steep tilings

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    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 Z2\mathbb{Z}^2 of the form 1xy21 \leq x-y \leq 2\ell for some integer 1\ell \geq 1, and are parametrized by a binary word w{+,}2w\in\{+,-\}^{2\ell} that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to w=(+)w=(+-)^\ell and to the limit case w=+w=+^\infty-^\infty. For each word ww 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

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    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

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    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

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    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 O(n(lnn)3)O(n (\ln n)^3) in approximate-size sampling and O(n4/3)O(n^{4/3}) 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 (a×b)(a\times b)-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

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    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
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