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15
J.Propp, The shape of a typical boxed plane partition
 J. of Math
, 1998
"... Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the ..."
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Cited by 57 (5 self)
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Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.
Kerov’s central limit theorem for the Plancherel measure on Young diagrams
, 2003
"... Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boun ..."
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Cited by 44 (7 self)
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Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan– Shepp 1977, Vershik–Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999.
A Grouptheoretic Approach to Fast Matrix Multiplication
 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science 2003, IEEE Computer Society
"... We develop a new, grouptheoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irr ..."
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Cited by 33 (4 self)
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We develop a new, grouptheoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n 2+o(1) support n × n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper. 1.
Gaussian Limit For The Plancherel Measure Of The Symmetric Group
 C. R. Acad. Sci. Paris
, 1993
"... this paper that the Plancherel measure of the symmetric group Sn converges weakly (as n !1) to a Gaussian random process in the infinite dimensional vector space. With respect to the law of large numbers for this group, found earlier in [1,2] this assertion is analogous to the central limit theorem. ..."
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Cited by 32 (3 self)
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this paper that the Plancherel measure of the symmetric group Sn converges weakly (as n !1) to a Gaussian random process in the infinite dimensional vector space. With respect to the law of large numbers for this group, found earlier in [1,2] this assertion is analogous to the central limit theorem. The idea of such a result was mentioned in [3]. In the course of preparation of the article I had numerous valuable conversations with A.Vershik and G.Olshanski. In particular, it was Olshanski who suggested to use Lemma 2.3 below while discussing the preliminary version of the paper.
The LowTemperature Expansion Of The Wulff Crystal In The 3D Ising Model
 Comm. Math. Phys
, 2001
"... We compute the expansion of the surface tension of the 3D random cluster model for q 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n goes to 1. This same shape determines the asymptotic Wulff crystal in the 3D Ising model (and more generally in ..."
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Cited by 26 (1 self)
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We compute the expansion of the surface tension of the 3D random cluster model for q 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n goes to 1. This same shape determines the asymptotic Wulff crystal in the 3D Ising model (and more generally in the 3D random cluster model for q 1) as the temperature goes to 0. 1.
Stochastic dynamics related to Plancherel measure
 AMS Transl.: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics
"... Dedicated to A. M. Vershik on the occasion of his 70th birthday Abstract. Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u, v) of this quadrant take the Young diagram obtained by applying the Robinson–Schensted correspondence to the intersectio ..."
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Cited by 14 (7 self)
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Dedicated to A. M. Vershik on the occasion of his 70th birthday Abstract. Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u, v) of this quadrant take the Young diagram obtained by applying the Robinson–Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0, 0), (u, 0), (u, v), (0, v). It is known that the distribution of the random Young diagram thus obtained is the poissonized Plancherel measure with parameter uv. We show that for (u, v) moving along any southeast–directed curve C in the quadrant, these Young diagrams form a Markov process ΛC with continuous time. We also describe ΛC in terms of jump rates. Our main result is the computation of the dynamical correlation functions of such Markov processes and their bulk and edge scaling limits.
Numerical Investigation of the Spectrum for Certain Families of Cayley Graphs
 in DIMACS Series in Disc. Math. and Theor. Comp. Sci
, 1993
"... In this paper we extend some earlier computations [8]. In particular, the expanding behavior of Cayley graphs of PSL2 (F107) is compared with that of the Cayley graphs for the group A10 . These computations support the (up to now) unvoiced conjecture of Lubotzky that the symmetric groups and project ..."
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Cited by 9 (5 self)
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In this paper we extend some earlier computations [8]. In particular, the expanding behavior of Cayley graphs of PSL2 (F107) is compared with that of the Cayley graphs for the group A10 . These computations support the (up to now) unvoiced conjecture of Lubotzky that the symmetric groups and projective linear groups have asymptotically different average expanding behavior. We also give a thorough spectral analysis for a natural family of Cayley graphs which does not admit analysis by Selberg's theorem. 1 Introduction Spectral analysis and operator theory have provided some of the main tools for the recent advances in constructions of expander graphs. In particular, by exploiting the various relationships between the second largest eigenvalue of the Laplacian and the expansion coefficient of graphs, families of expanders have been constructed and analyzed. When the graphs of interest are Cayley graphs, techniques from Fourier analysis are especially useful in this analysis. In this pap...
Large Deviations for Integer Partitions
, 1998
"... We consider deviations from limit shape induced by uniformly distributed partitions (and strict partitions) of an integer n on the associated Young diagrams. We prove a full large deviation principle, of speed p n. The proof, based on projective limits, uses the representation of the uniform measure ..."
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Cited by 9 (1 self)
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We consider deviations from limit shape induced by uniformly distributed partitions (and strict partitions) of an integer n on the associated Young diagrams. We prove a full large deviation principle, of speed p n. The proof, based on projective limits, uses the representation of the uniform measure on partitions by means of suitably conditioned independent variables.
PoissonDirichlet distribution for random Belyi surfaces
 Ann. Probab
, 2006
"... Abstract. Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a “typical” compact Riemann surface of large genus based on compactifying finitearea Riemann surfaces associated ..."
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Cited by 5 (0 self)
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Abstract. Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a “typical” compact Riemann surface of large genus based on compactifying finitearea Riemann surfaces associated with random cubic graphs; by a theorem of Belyi these are “dense ” in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichmüller spaces depends on the study of oriented cycles in random cubic graphs with random orientation; Brooks and Makover conjectured that asymptotically normalized cycles lengths follow PoissonDirichlet distribution. We present a proof of this conjecture using representation theory of the symmetric group. Consequently we also make progress towards a conjecture of Pippenger and Schleich which arose in the study of topological characteristics of random surfaces generated by cubic interactions. 1.
Central limit theorem for random partitions under the Plancherel measure, preprint ArXiv.org
, 2006
"... To the memory of Sergei Kerov (1946–2000) ..."
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