Hydrodynamic limit of the Robinson–Schensted–Knuth algorithm

Abstract

We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.

Poisson limit of bumping routes in the Robinson–Schensted correspondence

“…7 as the dashed line. Somewhat surprisingly it coincides with the hyperbola (9) shown in the projective coordinate system; a posteriori this gives some justification to the naive discussion from Sect. 1.8.…”

“…The following result shows a direct link between the above problem and the asymptotics of bumping routes. This result also shows an interesting link between the papers [9,14]. (34)…”

“…to be the position of the box containing ∞ in the appropriate insertion tableau. This problem was formulated by Duzhin [3]; the first asymptotic results in the scaling in which m → ∞ and t = O(m) were found by the first named author [9]. In the current paper we go beyond this scaling and consider m → ∞ and t = O m 2 ; the answer for this problem is essentially contained in Theorem 4.3.…”

“…Eq. (9) the Cartesian coordinates (x, y) of magnitude x, y = O √ m , the shape of the bumping route (scaled down by the factor 1 √ m ) converges in probability towards an explicit curve, which we refer to as the limit bumping curve, see Fig. 3 for an illustration.…”

“…This would correspond to the investigation of the asymptotics of the (non-rescaled) limit bumping curve x(y), y in the regime y → ∞. The latter analysis was performed by the first named author [9]; one of his results is that lim y→∞ x(y)y = 2; in other words, for y → ∞ the non-rescaled limit bumping curve can be approximated by the hyperbola x y = 2 while its rescaled counterpart which we consider in the current paper by the hyperbola…”