Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy_1 process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.Comment: 39 pages,6 figure

Finite time corrections in KPZ growth models

We consider some models in the Kardar-Parisi-Zhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion process. For these models, in the limit of large time t, universality of fluctuations has been previously obtained. In this paper we consider the convergence to the limiting distributions and determine the (non-universal) first order corrections, which turn out to be a non-random shift of order t^{-1/3} (of order 1 in microscopic units). Subtracting this deterministic correction, the convergence is then of order t^{-2/3}. We also determine the strength of asymmetry in the exclusion process for which the shift is zero. Finally, we discuss to what extend the discreteness of the model has an effect on the fitting functions.Comment: 34 pages, 5 figures, LaTeX; Improved version including shift of PASEP height functio

Polynuclear growth model, GOE2^2 and random matrix with deterministic source

We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE$^2$, which is defined as the square of the GOE Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE$^2$ as ``the square of GOE'', ours has an advantage that it can also describe the transition from the GUE Tracy-Widom distribution to the GOE$^2$. We further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain non-colliding Brownian motion model and the multi-layer PNG model with an external source. This provides us with a multi-matrix model interpretation of the multi-point height distributions of the PNG model with an external source.Comment: 27pages, 4 figure

Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers

The distribution function of the free energy fluctuations in one-dimensional directed polymers with $\delta$-correlated random potential is studied by mapping the replicated problem to the $N$-particle quantum boson system with attractive interactions. We find the full set of eigenfunctions and eigenvalues of this many-body system and perform the summation over the entire spectrum of excited states. It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.Comment: 23 page

Construction of a matrix product stationary state from solutions of finite size system

Stationary states of stochastic models, which have $N$ states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite $M$-dimensional matrix product state for any ${N,M}$. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and $N\le M$ are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where $M=N$ is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.Comment: 22 pages, no figure. Major changes: sec.3 was shortened; the list of references were changed. This is the final version, which will appear in J.Phys.

Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra

We study the partially asymmetric exclusion process with open boundaries. We generalise the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q. Due to the relationship between the matrix algebra and the q-deformed quantum harmonic oscillator algebra we find that q-Hermite polynomials, along with their orthogonality properties and generating functions, are of great utility. We employ two distinct sets of q-Hermite polynomials, one for q1. It turns out that these correspond to two distinct regimes: the previously studied case of forward bias (q1) where the boundaries support a current opposite in direction to the bulk bias. For the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size.Comment: 27 pages LaTeX2e, 3 figures, includes new references and further comparison with related work. To appear in J. Phys.

On the solvable multi-species reaction-diffusion processes

A family of one-dimensional multi-species reaction-diffusion processes on a lattice is introduced. It is shown that these processes are exactly solvable, provided a nonspectral matrix equation is satisfied. Some general remarks on the solutions to this equation, and some special solutions are given. The large-time behavior of the conditional probabilities of such systems are also investigated.Comment: 13 pages, LaTeX2

Systems Analysis and Structural Design of an Unpressurized Cargo Delivery Vehicle

The International Space Station will require a continuous supply of replacement parts for ongoing maintenance and repair after the planned retirement of the Space Shuttle in 2010. These parts are existing line-replaceable items collectively called Orbital Replacement Units, and include heavy and oversized items such as Control Moment Gyroscopes and stowed radiator arrays originally intended for delivery aboard the Space Shuttle. Current resupply spacecraft have limited to no capability to deliver these external logistics. In support of NASA's Exploration Systems Architecture Study, a team at Langley Research Center designed an Unpressurized Cargo Delivery Vehicle to deliver bulk cargo to the Space Station. The Unpressurized Cargo Delivery Vehicle was required to deliver at least 13,200 lbs of cargo mounted on at least 18 Flight Releasable Attachment Mechanisms. The Crew Launch Vehicle design recommended in the Exploration Systems Architecture Study would be used to launch one annual resupply flight to the International Space Station. The baseline vehicle design developed here has a cargo capacity of 16,000 lbs mounted on up to 20 Flight Releasable Attachment Mechanisms. Major vehicle components are a 5.5m-diameter cargo module containing two detachable cargo pallets with the payload, a Service Module to provide propulsion and power, and an aerodynamic nose cone. To reduce cost and risk, the Service Module is identical to the one used for the Crew Exploration Vehicle design

Will jams get worse when slow cars move over?

Motivated by an analogy with traffic, we simulate two species of particles (`vehicles'), moving stochastically in opposite directions on a two-lane ring road. Each species prefers one lane over the other, controlled by a parameter $0 \leq b \leq 1$ such that $b=0$ corresponds to random lane choice and $b=1$ to perfect `laning'. We find that the system displays one large cluster (`jam') whose size increases with $b$, contrary to intuition. Even more remarkably, the lane `charge' (a measure for the number of particles in their preferred lane) exhibits a region of negative response: even though vehicles experience a stronger preference for the `right' lane, more of them find themselves in the `wrong' one! For $b$ very close to 1, a sharp transition restores a homogeneous state. Various characteristics of the system are computed analytically, in good agreement with simulation data.Comment: 7 pages, 3 figures; to appear in Europhysics Letters (2005

Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries

The one-dimensional partially asymmetric simple exclusion process with open boundaries is considered. The stationary state, which is known to be constructed in a matrix product form, is studied by applying the theory of q-orthogonal polynomials. Using a formula of the q-Hermite polynomials, the average density profile is computed in the thermodynamic limit. The phase diagram for the correlation length, which was conjectured in the previous work[J. Phys. A {\bf 32} (1999) 7109], is confirmed.Comment: 24 pages, 6 figure