Abstract
We consider the Robinson–Schensted–Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the y-axis) when a specified number is inserted into a large Plancherel-distributed random tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.
Highlights
The set of Young diagrams will be denoted by Y; the set of Young diagrams with n boxes will be denoted by Yn
The set Y has a structure of an oriented graph, called Young graph; a pair μ λ forms an oriented edge in this graph if the Young diagram λ can be created from the Young diagram μ by addition of a single box
(17) states that the transition of the bumping route from the column x + 1 to the column x gives a multiplicative factor Rx to the total waiting time, with the factors R0, R1, . . . independent. It is more common in mathematical and physical models that the total waiting time for some event arises as a sum of some independent summands, so the multiplicative structure in Theorem 1.7 comes as a small surprise. We believe that this phenomenon can be explained heuristically as follows: when we study the transition of the bumping route from row y to the row y + 1, the probability of the transition from column x + 1 to column x seems asymptotically to be equal to x+1 +o 1 y y for fixed value of x, and for y → ∞
Summary
The set of Young diagrams will be denoted by Y; the set of Young diagrams with n boxes will be denoted by Yn. We will draw Young diagrams and tableaux in the French convention with the Cartesian coordinate system O x y, cf Figs. We index the rows and the columns of tableaux by non-negative integers from N0 = {0, 1, 2, . }. In particular, if is a box of a tableau, we identify it with the Cartesian coordinates of its lower-left corner:. For a tableau T we denote by Tx,y its entry which lies in the intersection of the row y ∈ N0 and the column x ∈ N0. The position of the box s in the tableau T will be denoted by Poss(T ) ∈ N0 × N0. ) are indexed by the elements of N0; in particular the length of the bottom row of λ is denoted by λ0
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