Rigorous mathematical models of lastpassage percolation study the evolution of a random cluster which grows in two dimensions by following some precise, predetermined rules. The speed of the growth depends on an underlying 'random environment' that is attached to the space, effectively creating a 'random medium'.
Theoretically, one can devise various mechanisms that explain how growth occurs, and in lastpassage percolation the most common one can be explained using a queues in series system (called a 'totally asymmetric simple exclusion process') in which many customers queue up in a single line in front of a series of servers. For the benefit of exposition, consider a concrete yet nightmarish situation in a small airport, where passengers queue in a single file for checkin, then luggagedrop, followed by passport control, security and a finally queue at the gate. Without cutting in front of other customers, all of the servers need to be cleared by everyone, in order. When the first customer leaves the first server, she goes immediately to the second one. The second customer then starts being serviced by server 1, while the second one serves customer 1 and the system keeps evolving. Service times are random and the question is when a certain customer (e.g. the nth) will clear a given server (e.g. the mth). This random time (where n clears m) is called the lastpassage time (for point (m,n)) and it is precisely the time that point (m,n) joins the random cluster. Every time a customer passes through a server, a corresponding point is added to the cluster. The random service times correspond to the random environment. This is the particle system interpretation of nearest neighbor lastpassage percolation. There is also the geometric interpretation: At any given time we can be view this as the evolution of a random surface and, as such, we are concerned with some welldefined geometric notions such as shape, random geodesics and curvature. Not surprisingly, the distribution of the random environment has an effect on the shape the surface has for large times.
In some examples, mathematicians can compute the shape explicitly and check that, when suitably scaled, it can be viewed as a 'strictly concave, differentiable' surface; in other words, if we 'zoom out' and look at the boundary of the random cluster from afar, we will see a nice curved shape with no flat segments (strict concavity) with no sharp points (differentiability). Projects in this proposal study this shape when it exhibits something called a `flat edge' , which says that the shape is not strictly concave anymore.
This is a theoretical mathematical proposal. We will study the lastpassage percolation shape in an 'inhomogeneous' environment. The word 'inhomogeneous' means that each server changes the way it serves people after some time, e.g. he can get tired, so he slows down, or becomes more efficient, so he speeds up. In one studied example the limiting shape function exhibits a flat edge and at least one point of nondifferentiability. We will extend these results not only to nearest neighbour directed lastpassage percolation but to a more general class of models, like a discrete version of the Hammersley process, which acts as an independent version of the harder 'longest common subsequence' problem. In the process, we will also develop the mathematical groundwork for the microscopic effects of the flat edge, in particular how it affects the geodesics' (special paths that can tell us where the delays happened in the particle system) of the model. Furthemore, we will establish a connection between the creation of the flat edge and macroscopic fluctuations of geodesics. Understanding this is crucial for many reasons, one of the most important being that this will provide further evidence that models with flat edges do not belong in the famous KPZ universality class.
