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Abstract : |
A bijection is given between fixed point free involutions of {1, 2,..., 2N} with maximum decreasing subsequence size 2p and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points l ? 1. In one class of walker configurations the maximum displacement of the right most walker is p. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker. Random permutations are fundamental combinatorial objects, which are intimately related to other fundamental combinatorial objects such as Young tableaux via the Robinson-Schensted-Knuth correspondence. We recall that a Young tableau can be regarded as a numbered diagram of a partition 1 ? 2 ? ? p ? 0. The diagram consists of squares drawn within a matrix array with a square drawn in each row (1 ? j ? p) and column k (1 ? k ? j), while in each square is recorded a number specified by some rule. Recently random permutations, Young tableaux and their generalizations have been shown to be at the core of certain statistical mechanical models of growth processes [16, 15, 19, 14], vicious walker paths [13, 10, 5, 17] and exclusion processes (the latter via mappings to certain growth processes and vicious walker paths) amongst other topics. This has led to progress in the study of these statistical mechanical models, by way of the progress in the determination of fluctuation formulas for quantities associated with random permutations [4, 2, 3]. As an example of the insight gained, we draw attention to the work of Prhoffer and Spohn [19]. These authors identify distinct scaling forms for growth models in the Kardar-Parisi-Zhang (KPZ) universality class, that is growth models described by the KPZ equation ?h, |
