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.

Abstract. 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 It is largely based on Kerov’s unpublished work notes,

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.

We develop a new, group-theoretic 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.

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.

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...

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.

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 finite-area 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 Poisson-Dirichlet 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.

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