Quantum Loewner Evolution

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} $\eta$-DBM, a generalization of DLA in which particle locations are sampled from the $\eta$-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $\eta$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $\gamma \in [0,2]$. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE$(\gamma^2, \eta)$. QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion $\nu_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of $\nu_t$ using an SPDE. For each $\gamma \in (0,2]$, there are two or three special values of $\eta$ for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of $\nu_t$. We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation. We propose QLE$(2,1)$ as a scaling limit for DLA on a random spanning-tree-decorated planar map, and QLE$(8/3,0)$ as a scaling limit for the Eden model on a random triangulation. We propose using QLE$(8/3,0)$ to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE$(8/3,0)$, up to a fixed time, as a metric ball in a random metric space.Comment: 132 pages, approximately 100 figures and computer simulation

Aerodynamics support of research instrument development

A new velocimetry system is currently being developed at NASA LaRC. The device, known as a Doppler global velocimeter (DGV), can record three velocity components within a plane simultaneously and in near real time. To make measurements the DGV, like many other velocimetry systems, relies on the scattering of light from numerous small particles in a flow field. The particles or seeds are illuminated by a sheet of laser light and viewed by two CCD cameras. The scattered light from the particles will have a frequency which is a function of the source laser light frequency, the viewing angle, and most importantly the seed velocities. By determining the scattered light intensity the velocity can be measured at all points within the light sheet simultaneously. Upon completion of DGV component construction and initial check out a series of tests in the Basic Aerodynamic Research (wind) Tunnel (BART) are scheduled to verify instrument operation and accuracy. If the results are satisfactory, application of the DGV to flight measurements on the F-18 High Alpha Research Vehicle (HARV) are planned. The DGV verification test in the BART facility will utilize a 75 degree swept delta wing model. A major task undertaken this summer included evaluation of previous results for this model. A specific series of tests matching exactly the previous tests and exploring new DGV capabilities were developed and suggested. Another task undertaken was to study DGV system installation possibilities in the F-18 HARV aircraft. In addition, a simple seeding system modification was developed and utilized to make Particle Imaging Velocimetry (PIV) measurements in the BART facility

An evaluation of preliminary Doppler global velocimetry measurements

A review of Doppler Global Velocimetry (DGV) data obtained during wind tunnel tests on a 75 degree swept delta wing was performed. High frequency variations observed in normalized data files are attributed to image alignment problems. Unfortunately, initial DGV velocity data compared poorly with baseline reference data. Nonlinear DGV system operation during the tests is the likely source of this problem. Corrected data compares much more favorably and suggests that DGV is a valid measurement technique. Future DGV investigations should include a method or means for monitoring laser frequency relative to the ALF transfer function behavior

Slow modes in Keplerian disks

Low-mass disks orbiting a massive body can support "slow" normal modes, in which the eigenfrequency is much less than the orbital frequency. Slow modes are lopsided, i.e., the azimuthal wavenumber m=1. We investigate the properties of slow modes, using softened self-gravity as a simple model for collective effects in the disk. We employ both the WKB approximation and numerical solutions of the linear eigenvalue equation. We find that all slow modes are stable. Discrete slow modes can be divided into two types, which we label g-modes and p-modes. The g-modes involve long leading and long trailing waves, have properties determined by the self-gravity of the disk, and are only present in narrow rings or in disks where the precession rate is dominated by an external potential. In contrast, the properties of p-modes are determined by the interplay of self-gravity and other collective effects. P-modes involve both long and short waves, and in the WKB approximation appear in degenerate leading/trailing pairs. Disks support a finite number---sometimes zero---of discrete slow modes, and a continuum of singular modes.Comment: 32 pages, 12 figures. To be published in Astronomical Journa

Inverse Magnetic Catalysis in Bottom-Up Holographic QCD

We explore the effect of magnetic field on chiral condensation in QCD via a simple bottom up holographic model which inputs QCD dynamics through the running of the anomalous dimension of the quark bilinear. Bottom up holography is a form of effective field theory and we use it to explore the dependence on the coefficients of the two lowest order terms linking the magnetic field and the quark condensate. In the massless theory, we identify a region of parameter space where magnetic catalysis occurs at zero temperature but inverse magnetic catalysis at temperatures of order the thermal phase transition. The model shows similar non-monotonic behaviour in the condensate with B at intermediate T as the lattice data. This behaviour is due to the separation of the meson melting and chiral transitions in the holographic framework. The introduction of quark mass raises the scale of B where inverse catalysis takes over from catalysis until the inverse catalysis lies outside the regime of validity of the effective description leaving just catalysis.Comment: 9 pages, 8 figure

Spatio-temporal Modelling of Remote-sensing Lake Surface Water Temperature Data

Remote-sensing technology is widely used in environmental monitoring. The coverage and resolution of satellite based data provide scientists with great opportunities to study and understand environmental change. However, the large volume and the missing observations in the remote-sensing data present challenges to statistical analysis. This paper investigates two approaches to the spatio-temporal modelling of remote-sensing lake surface water temperature data. Both methods use the state space framework, but with different parameterizations to reflect different aspects of the problem. The appropriateness of the methods for identifying spatial/temporal patterns in the data is discussed

Functional PCA for Remotely Sensed Lake Surface Water Temperature Data

Functional principal component analysis is used to investigate a high-dimensional surface water temperature data set of Lake Victoria, which has been produced in the ARC-Lake project. Two different perspectives are adopted in the analysis: modelling temperature curves (univariate functions) and temperature surfaces (bivariate functions). The latter proves to be a better approach in the sense of both dimension reduction and pattern detection. Computational details and some results from an application to Lake Victoria data are presented

Within Lake Clustering of High Resolution Satellite Retrievals: A Functional Data and Clustering Approach

No abstract available