Limit processes for TASEP with shocks and rarefaction fans

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.Comment: 46 pages, 3 figures, LaTeX; Extended explanations in the first two section

Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [Algebr. Geom. Topol. 7 (2007):1297-1326]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page

Universality of slow decorrelation in KPZ growth

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent $z=3/2$, that means one should find a universal space-time limiting process under the scaling of time as $t\,T$, space like $t^{2/3} X$ and fluctuations like $t^{1/3}$ as $t\to\infty$. In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times $t\, T$ and $t\, T+t^{\nu}$ are identical, for any $\nu<1$. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally / partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to space-time regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applied to a wide variety of models.Comment: Exposition improved, typos correcte

The American Religious Landscape and the 2004 Presidential Vote: Increased Polarization

Presents findings from a post-election survey conducted in November and December 2004. Explores the polarization between different religions, as well as within the major religious traditions

A quasi-pure Bose-Einstein condensate immersed in a Fermi sea

We report the observation of co-existing Bose-Einstein condensate and Fermi gas in a magnetic trap. With a very small fraction of thermal atoms, the 7Li condensate is quasi-pure and in thermal contact with a 6Li Fermi gas. The lowest common temperature is 0.28 muK = 0.2(1) T_C = 0.2(1) T_F where T_C is the BEC critical temperature and T_F the Fermi temperature. Behaving as an ideal gas in the radial trap dimension, the condensate is one-dimensional.Comment: 4 pages, 5 figure

Height fluctuations for the stationary KPZ equation

We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X), for B(X) a two-sided standard Brownian motion) and show that as time T goes to infinity, the fluctuations of the height function H(T,X) grow like T^{1/3} and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift beta to the left of the origin and b to the right of the origin. The Fredholm determinant has a limit for beta>b, and the case where beta=b (corresponding to the stationary initial data) follows from an analytic continuation argument.Comment: 91 pages, 8 figure

Modeling Course-Based Undergraduate Research Experiences: An Agenda for Future Research and Evaluation

Course-based undergraduate research experiences (CUREs) are being championed as scalable ways of involving undergraduates in science research. Studies of CUREs have shown that participating students achieve many of the same outcomes as students who complete research internships. However, CUREs vary widely in their design and implementation, and aspects of CUREs that are necessary and sufficient to achieve desired student outcomes have not been elucidated. To guide future research aimed at understanding the causal mechanisms underlying CURE efficacy, we used a systems approach to generate pathway models representing hypotheses of how CURE outcomes are achieved. We started by reviewing studies of CUREs and research internships to generate a comprehensive set of outcomes of research experiences, determining the level of evidence supporting each outcome. We then used this body of research and drew from learning theory to hypothesize connections between what students do during CUREs and the outcomes that have the best empirical support. We offer these models as hypotheses for the CURE community to test, revise, elaborate, or refute. We also cite instruments that are ready to use in CURE assessment and note gaps for which instruments need to be developed.Howard Hughes Medical InstituteScience and Mathematics Educatio

Thermodynamic phase-field model for microstructure with multiple components and phases: the possibility of metastable phases

A diffuse-interface model for microstructure with an arbitrary number of components and phases was developed from basic thermodynamic and kinetic principles and formalized within a variational framework. The model includes a composition gradient energy to capture solute trapping, and is therefore suited for studying phenomena where the width of the interface plays an important role. Derivation of the inhomogeneous free energy functional from a Taylor expansion of homogeneous free energy reveals how the interfacial properties of each component and phase may be specified under a mass constraint. A diffusion potential for components was defined away from the dilute solution limit, and a multi-obstacle barrier function was used to constrain phase fractions. The model was used to simulate solidification via nucleation, premelting at phase boundaries and triple junctions, the intrinsic instability of small particles, and solutal melting resulting from differing diffusivities in solid and liquid. The shape of metastable free energy surfaces is found to play an important role in microstructure evolution and may explain why some systems premelt at phase boundaries and phase triple junctions while others do not.Comment: 14 pages, 8 figure

From interacting particle systems to random matrices

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of hermitian random matrices, but this analogy does not extend to the case of symmetric matrices. Therefore the connections between growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor corrections in scaling of section 2.