Maximal Cells in Shifted Staircase Tableaux and a Quarter-Circle Law

In this note, we explicitly compute the probability that a given cell in a random standard Young tableau of the shifted staircase shape $(2n-1, 2n-3, \ldots, 3,1)$ contains the maximal label. We also show that the asymptotic distribution of the cell containing the maximal label is governed by the quarter-circle law. The bijection between the tableaux and thereduced decompositions of the longest element of the group $B_n$ of the signed permutations yields the probability distribution of the first (and any) letter of the random reduced decompositions. We also show the results of some computational experiments on the random sorting networks of $B_n$.