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 non-differentiability, at segments, and non-uniqueness 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 non-boundary 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 Burke-type 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 log-moment generating functions in those directions.

The study was conducted on the New River sector of the Radford Arsenal, a 2322-acre enclosure in Pulaski County, Virginia. Seventy-five per cent of the enclosed area is open to grassland and the entire area is interlaced with roads. The combination of the openness of terrain and easy accessibility to nearly all portions made observation of deer relatively easy. The 1966 pre-season deer herd estimated by means of observations and the Lincoln index, at 334, with a composition of 25.4% adult bucks, 44.3% adult does, and 30.3% fawns. The 1966 reproductive rate was determined to be 1.1 fawns per breeding female. Females were assumed to first produce fawns at 2 years of age. Using known data and assumptions based on findings on similar areas, a hypothetical growth table for the herd was compiled to present one way in which the herd might have grown since the first deer were introduced in 1952. Bow hunting has been allowed on the area since 1959. During 8 years of hunting, 1836 bow hunters have taken 172 deer for an average hunter success of 9.3%. In 1966, 63 deer were removed by hunters and 22 known cripple losses occurred. The home range size was determined for 36 fawns, 8 yearlings, and 2 adults. The 24 male fawn home ranges averaged 125 acres (34-371 acres), while the 12 female fawn home ranges averaged 133 acres (34-273 acres). The home range of the 7 yearling females averaged 172 acres (96-357 acres), and the 1 yearling male has a range of 155 acres in size. The single adult male had a home range of 120 acres while the 1 adult female’s home range was 432 acres. Thirty-one fawns, 21 males and 10 females, and 2 yearling females were observed to move outside of their home ranges when pressed by hunters. The distance moved by the fawns from .3 mile to 2.0 miles from the center of activity of their home ranges. The yearling females moved .4 and 1.7 mile from their centers of activity. All of these deer returned to their original home range when the hunting pressure was removed. / Master of Science