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.

This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts

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.

this paper is to show that such a phenomenon holds for more general partitions

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τn = τn(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τn − c0n 2/3)/c1n 1/3 log 1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c0 = ζ(2)/[2ζ(3)] 2/3, c1 = √ 2/3/[2ζ(3)] 1/3 and ζ(s) = ∑ ∞ j=1 j−s.

This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in partitions of n. It contains also formulas for the moments (under uniform distribution) of k th parts in partitions of n. 1.

For a subfamily of multiplicative measures on integer partitions we give conditions for associated Young diagrams to converge in probability after a proper rescaling to a certain curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.

1. In what follows we are dealing with some statistical properties of partitions resp. unequal partitions of positive integers. We introduce the notation (1.1) for a generic partition n of n where 11.2) m = mfl7) and the i,‘s are integers. Let p(n) denote the number of partitions

Considering the conjugacy classes of the alternating group of degree n, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is I- e(n), and e(n)- 0 as n

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