A note on limit shapes of minimal difference partitions

We provide a variational derivation of the limit shape of minimal difference partitions and discuss the link with exclusion statistics. This paper is dedicated to Professor Leonid Pastur for his 70th anniversary. A partition of a natural integer E [1] is a decomposition of E as a sum of a nonincreasing sequence of positive integers {hj}, i.e., E = ∑ j hj such that hj ≥ hj+1, for j = 1, 2 . . .. For example, 4 can be partitioned in 5 ways: 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. Partitions can be graphically represented by Young diagrams (also called Ferrers diagrams) where hj corresponds to the height of the j-th column. The {hj}’s are called the parts or the summands of the partition. One can put several constraints on such partitions. For example, one can take the number of columns N to be fixed or put restrictions on the heights. In this paper we focus on a particular constrained partition problem called the minimal difference p partitions (MDP–p). The MDP–p problem is defined by restricting the height difference between two neighboring columns, hj−hj+1 ≥ p. For instance the only allowed partitions of 4 with p = 1 are 4 and 3+1. A typical Young diagram for MDP–p problem is shown in figure 1. Consider now the set of all possible partitions of E satisfying E = ∑ j hj and hj − hj+1 ≥ p. Since this is a finite set, one can put a uniform probability measure on it, which means that all partitions are equiprobable. Then, a natural question is: what is the typical shape of a Young diagram when E → ∞? In the physics literature this problem was first raised by Temperley, who was interested in determining the equilibrium profile of a simple cubic crystal grown from the corner of three walls at right angles. The two dimensional version of the problem —where walls (two) are along the horizontal and the vertical axes and E “bricks” (molecules) are packed into the first quadrant one by one such that each brick, when it is added, makes two contact along faces— corresponds