The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann--Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to factorize into two independent Airy processes using the Deift-Zhou steepest descent analysis. Additionally we relate the theory of Fredholm determinants of integrable kernels and the theory of isomonodromic tau function. Using the Riemann-Hilbert problem mentioned above we construct a suitable Lax pair formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs recently found and additionally find a third one not reducible to those.Comment: 43 pages, 7 figures. Final version with minor changes. Accepted for publication on International Mathematical Research Notice

Parton cascade and coalescence

This is a review of the parton cascade approach and its implications on parton coalescence at RHIC.Comment: Invited plenary talk at Quark Matter 2005 (Aug 4-9, 2005, Budapest, Hungary) - to appear in proceedings. 10 pages, 12 EPS figures, ESP style file include

Empirical Constraints on Parton Energy Loss in Nucleus-Nucleus Collisions at RHIC

We present empirical features of parton energy loss in nucleus-nucleus collisions at RHIC through studies of the spectra and nuclear modification factors ($R_{AA}$) for charged hadrons, neutral pions ($\pi^0$) and non-photonic electrons. The flat distribution of $R_{AA}$ at high transverse momentum ($p_{T}$) for a given collision centrality is consistent with a scenario where parton energy loss $\Delta p_{T}$ is proportional to $p_{T}$. The centrality dependence of the parton energy loss indicates the absence of path length dependence in the magnitude of energy loss. The lack of strong path length dependence suggests a dynamical picture where the dense partonic medium undergoes rapid expansion and the density of the medium falls rapidly in the first a few Fermi interval, which may be much shorter than the full path length. Implications of the empirical constraints on the parton energy loss will also be discussed.Comment: 6 pages, 5 figures, submitted to Phys. Lett.

Grassmann geometries in infinite dimensional homogeneous spaces and an application to reflective submanifolds

Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. The solutions are shown to correspond to various special submanifolds, depending on which homogeneous space U/L one projects to. We apply the construction to a question which generalizes, to the context of reflective submanifolds of arbitrary symmetric spaces, the problem of isometric immersions of space forms with negative extrinsic curvature and flat normal bundle. For this problem, we prove that the only cases where local solutions exist are the previously known cases of space forms, in addition to constant curvature Lagrangian immersions into complex projective and complex hyperbolic spaces. We also prove non-existence of global solutions in the compact case. The solutions associated to other reflective submanifolds correspond to special deformations of lower dimensional submanifolds. As an example, we obtain a special class of surfaces in the 6-sphere.Comment: 31 pages. Minor revision. Some notational changes, comments added. Section 6.5 has been added. Section 8.1 rewritte

Radiative and Collisional Energy Loss, and Photon-Tagged Jets at RHIC

The suppression of single jets at high transverse momenta in a quark-gluon plasma is studied at RHIC energies, and the additional information provided by a photon tag is included. The energy loss of hard jets traversing through the medium is evaluated in the AMY formalism, by consistently taking into account the contributions from radiative events and from elastic collisions at leading order in the coupling. The strongly-interacting medium in these collisions is modelled with (3+1)-dimensional ideal relativistic hydrodynamics. Putting these ingredients together with a complete set of photon-production processes, we present a calculation of the nuclear modification of single jets and photon-tagged jets at RHIC.Comment: 4 pages, 4 figures, contributed to the 3rd International Conference on Hard and Electro-Magnetic Probes of High-Energy Nuclear Collisions (Hard Probes 2008), typos corrected, published versio

Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains

A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlev\'e equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.Comment: latex2e, 41 pages, no figure; (v2) some minor errors are corrected; (v3) fully revised and shortend, and some results are improve

The power of choice in network growth

The "power of choice" has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k=2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k >> 1 to see a power law over a wide range of degrees.Comment: 9 pages, 4 figure

Collisional energy loss and the suppression of high pTp_T hadrons

We calculate nuclear suppression factor ($R_{AA}$) for light hadrons by taking only the elastic processes and argue that in the measured $p_T$ domain of RHIC, collisional rather than the radiative processes is the dominant mechanism for partonic energy loss.Comment: Presented at the international conference on strong and electroweak matter 2006, May 10-13, Brookhaven National Laborator