On Asymptotics for the Airy Process

The Airy process A(t), introduced by Pr\"ahofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s_1, s_2, and t for the probability that A(0)<s_1 and A(t)<s_2. Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t->infinity, with fixed s_1 and s_2. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are formulas obtained by the author and C. A. Tracy in the derivation of the Painlev\'e II representation for the distribution function F_2 plus a few others obtained in the same way.Comment: 5 pages, LaTex fil

Some Classes of Solutions to the Toda Lattice Hierarchy

We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and Fredholm determinants. Others are converted into a class of operator solutions to the $l$-periodic Toda hierarchy.Comment: LaTeX file, 18 pages. Results generalized and applications to the Toda equations adde

On the Eigenvalues of Certain Canonical Higher-Order Ordinary Differential Operators

We consider the operator of taking the $2p$th derivative of a function with zero boundary conditions for the function and its first $p-1$ derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for $p=1$ and $p=3$.Comment: LaTeX, 12 pages, 2 figure