Abstract: The Preisach graph is a directed graph associated with a permutation ρ∈SN. We give an explicit bijection between the vertices and increasing subsequences of ρ, with the property that its length equals the degree of nesting of the vertex inside a hierarchy of cycles and sub-cycles. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.

Abstract: We consider the asymmetric simple exclusion process on Z with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X(t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X(t) is asymptotically distributed as the difference of two independent, GUE-distributed random variables.The key part of the proof is to show that X(t) equals, up to a negligible term, the difference of a random number of holes and particles, with the randomness built up by ASEP itself. This provides a KPZ analogue to the 1994 result of Ferrari and Fontes, where this randomness comes from the initial data and leads to Gaussian limit laws.

Abstract: In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik–Rains distributions for full-space geometry. The result is obtained by using a related integrable model, having Pfaffian structure, together with careful analytic continuation and steepest descent analysis.