We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities that they li
e below a given curve as Fredholm determinants of certain boundary value operators, and the several applications of these formulas to variational problems involving Airy processes that arise in physical problems, as well as to their local behaviour.
We consider one component lattice gases with a local dynamics and a stationary product Bernoulli measure. We give upper and lower bounds on the diffusivity at an equilibrium point depending on the dimension and the local behavior of the macroscopic f
lux function. We show that if the model is expected to be diffusive, it is indeed diffusive, and, if it is expected to be superdiffusive, it is indeed superdiffusive.
We prove that the random variable $ct=argmax_{tinrr}{aip(t)-t^2}$ has tails which decay like $e^{-ct^3}$. The distribution of $ct$ is a universal distribution which governs the rescaled endpoint of directed polymers in 1+1 dimensions for large time or temperature.
The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp{a}lainen and Balazs recently proved this conjecture for the nearest neighbor case. We extend their resu
lts to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.
