Random walks and random fixed-point free involutions

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.

Fermionic construction of tau functions and random processes

“…To study the phenomenon of shock waves and phase transition of the first order known for the problem with open boundaries [55], [58] (6) to study the phenomenon of thermalization which should occur for finite graphs (7) to understand links with other approaches to the 1D non-equilibrium phenomena, in particular with these of papers [21], [22], [59], [57]. (8) to understand relations with the Bethe anzats method for the ASEP model reactions [57] and S.I.Badulin for technical support. J. H. would like to thank Herbert Spohn for helpful discussions in Nov. -Dec. 2006, at the Kavli Institute for Theoretical Physics, U.C.…”

“…The result of the application of o(T) := o T,T−1 · · · o 1,0 (7) to the initial state ν (corresponding to an initial configuration of hard core particles), is a Fock vector which is a linear combination of basis Fock vectors (=configurations of hard core particles). One of the objects of an interest is the relative weight of a given configuration, say, λ, W ν→λ (T) = λ|o(T)|ν (8) with respect to the weight of all possible configurations,…”

Osculating Paths and Oscillating Tableaux

“…An example of an element of OP(4, 6, {1, 2, 3}, {1, 4, 5}, 11) is P = (4, 1), (3, 1), (3, 2), (3,3), (3,4), (2,4), (2,5), (1,5), (1,6) , (4,4), (3,4), (3,5), (2,5), (2,6) , (4,5), (4,6), (3,6) , which is shown diagrammatically in Figure 2.…”

Tiling bijections between paths and Brauer diagrams

“…For information on bijections between Brauer diagrams (and more general partitions) and tableaux and pairs of walks, we refer to [6,12,14,17,23,24,26]. We also remark that the article [2] gives a bijection between fixed-point free involutions of a set of size 2n and certain sets of tuples of non-intersecting walks on the natural numbers arising in statistical mechanics (the random-turns model of vicious random walkers).…”

“…Standard tableaux of shape ((2),(2)) and the corresponding walksProposition [18] Let λ ∈ Γ d n and assume that [h x i,i+1 (T )] = 0 for all i ∈ {1, 2, . .…”