Order of the variance in the discrete Hammersley process

We discuss the order of the variance on a lattice analogue of the Hammersley process, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope.

For this model the shape function exhibits two flat edges and we study the order of the variance in directions that fall in the flat edge, in directions that approximate the edge of the flat edge, and in directions in the strictly concave section of the shape for the i.i.d. model and for the associated equilibrium model with boundaries. If time permitting, we will discuss the shape function and variance in some inhomogeneous models as well.

This is an exposition of several works with Janosch Ortmann, Elnur Emrah and Federico Ciech.