Title:

Models of last passage percolation

The thesis provides the discussion of three last passage percolation models. In particular, we focus on three aspects of probability theory: the law of large numbers, the order of the variance and large deviation estimates. In Chapter 1, we give a brief introduction to the percolation models in general and we present some important results for this topic which are heavily used in the following proofs. In Chapter 2, we prove a strong law of large numbers for directed last passage times in an independent but inhomogeneous exponential environment. Rates for the exponential random variables are obtained from a discretisation of a speed function that may be discontinuous on a locally finite set of discontinuity curves. The limiting shape is cast as a variational formula that maximises a certain functional over a set of weakly increasing curves. Using this result, we present two examples that allow for partial analytical tractability and show that the shape function may not be strictly concave, and it may exhibit points of nondifferentiability, at segments, and nonuniqueness of the optimisers of the variational formula. Finally, in a specific example, we analyse further the macroscopic optimisers and uncover a phase transition for their behaviour. In Chapter 3, we discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, 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. We show that along characteristic directions the order of the variance of the last passage time is of order N2=3 in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N) in the boundary model and to O(1) for the nonboundary model. This behavior is the result of the two at edges of the shape function. In Chapter 4, we prove a large deviation principle and give an expression for the rate function, for the last passage time in a Bernoulli environment. The model is exactly solvable and its invariant version satisfies a Burketype property. Finally, we compute explicit limiting logarithmic moment generating functions for both the classical and the invariant models. The shape function of this model exhibits a flat edge in certain directions, and we also discuss the rate function and limiting logmoment generating functions in those directions.
