Central limit theorems for the spectra of classes of random fractals

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known

Transition density of diffusion on Sierpinski gasket and extension of Flory's formula

Some problems related to the transition density u(t,x) of the diffusion on the Sierpinski gasket are considerd, based on recent rigorous results and detailed numerical calculations. The main contents are an extension of Flory's formula for the end-to-end distance exponent of self-avoiding walks on the fractal spaces, and an evidence of the oscillatory behavior of u(t,x) on the Sierpinski gasket.Comment: 11 pages, REVTEX, 2 postscript figure

The WiggleZ Dark Energy Survey: Survey Design and First Data Release

The WiggleZ Dark Energy Survey is a survey of 240,000 emission line galaxies in the distant universe, measured with the AAOmega spectrograph on the 3.9-m Anglo-Australian Telescope (AAT). The target galaxies are selected using ultraviolet photometry from the GALEX satellite, with a flux limit of NUV<22.8 mag. The redshift range containing 90% of the galaxies is 0.2<z<1.0. The primary aim of the survey is to precisely measure the scale of baryon acoustic oscillations (BAO) imprinted on the spatial distribution of these galaxies at look-back times of 4-8 Gyrs. Detailed forecasts indicate the survey will measure the BAO scale to better than 2% and the tangential and radial acoustic wave scales to approximately 3% and 5%, respectively. This paper provides a detailed description of the survey and its design, as well as the spectroscopic observations, data reduction, and redshift measurement techniques employed. It also presents an analysis of the properties of the target galaxies, including emission line diagnostics which show that they are mostly extreme starburst galaxies, and Hubble Space Telescope images, which show they contain a high fraction of interacting or distorted systems. In conjunction with this paper, we make a public data release of data for the first 100,000 galaxies measured for the project.Comment: Accepted by MNRAS; this has some figures in low resolution format. Full resolution PDF version (7MB) available at http://www.physics.uq.edu.au/people/mjd/pub/wigglez1.pdf The WiggleZ home page is at http://wigglez.swin.edu.au

Serum Amyloid A Stimulates Vascular and Renal Dysfunction in Apolipoprotein E-Deficient Mice Fed a Normal Chow Diet

Elevated serum amyloid A (SAA) levels may promote endothelial dysfunction, which is linked to cardiovascular and renal pathologies. We investigated the effect of SAA on vascular and renal function in apolipoprotein E-deficient (ApoE−/−) mice. Male ApoE−/− mice received vehicle (control), low-level lipopolysaccharide (LPS), or recombinant human SAA by i.p. injection every third day for 2 weeks. Heart, aorta and kidney were harvested between 3 days and 18 weeks after treatment. SAA administration increased vascular cell adhesion molecule (VCAM)-1 expression and circulating monocyte chemotactic protein (MCP)-1 and decreased aortic cyclic guanosine monophosphate (cGMP), consistent with SAA inhibiting nitric oxide bioactivity. In addition, binding of labeled leukocytes to excised aorta increased as monitored using an ex vivo leukocyte adhesion assay. Renal injury was evident 4 weeks after commencement of SAA treatment, manifesting as increased plasma urea, urinary protein, oxidized lipids, urinary kidney injury molecule (KIM)-1 and multiple cytokines and chemokines in kidney tissue, relative to controls. Phosphorylation of nuclear-factor-kappa-beta (NFκB-p-P65), tissue factor (TF), and macrophage recruitment increased in kidneys from ApoE−/− mice 4 weeks after SAA treatment, confirming that SAA elicited a pro-inflammatory and pro-thrombotic phenotype. These data indicate that SAA impairs endothelial and renal function in ApoE−/− mice in the absence of a high-fat diet

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

Spectral asymptotics for stable trees

We calculate the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on alpha-stable trees, which lead in turn to short-time heat kernel asymptotics for these random structures. In particular, the conclusions we obtain demonstrate that the spectral dimension of an alpha-stable tree is almost-surely equal to 2 alpha/(2 alpha-1), matching that of certain related discrete models. We also show that the exponent for the second term in the asymptotic expansion of the eigenvalue counting function is no greater than 1/(2 alpha-1). To prove our results, we adapt a self-similar fractal argument previously applied to the continuum random tree, replacing the decomposition of the continuum tree at the branch point of three suitably chosen vertices with a recently developed spinal decomposition for alpha-stable trees

Convergence of mixing times for sequences of random walks on finite graphs

We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdos-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal graphs, Galton-Watson trees and the range of a high-dimensional random walk

Random fractal dendrites

Dendrites are tree-like topological spaces, and in this thesis, the physical characteristics of various random fractal versions of this type of set are investigated. This work will contribute to the development of analysis on fractals, an area which has grown considerably over the last twenty years. First, a collection of random self-similar dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have often relied on the scaling factors being bounded uniformly away from zero. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that this condition is not necessary; a simple condition on the tail of the distribution of the scaling factors at zero is all that is assumed. The scaling factors of these recursively defined structures form what is known as a multiplicative cascade, and results about the height of this random object are also obtained. With important physical and probabilistic applications, the heat equation has justifiably received a substantial amount of attention in a variety of settings. For certain types of fractals, it has become clear that a key factor in estimating the heat kernel is the volume growth with respect to the resistance metric on the space. In particular, uniform polynomial volume growth, which occurs for many deterministic self-similar fractals, immediately implies uniform (on-diagonal) heat kernel behaviour. However, in the random fractal setting, this is frequently not the case, and volume fluctuations are often observed. Motivated by this, an analysis of how volume fluctuations lead to corresponding heat kernel fluctuations for measure-metric spaces equipped with a resistance form is conducted here. These results apply to the aforementioned random self-similar dendrites, amongst other examples. The continuum random tree (CRT) of Aldous is an important random example of a measure-metric space, and fits naturally into the framework of the previous paragraph. In this thesis, quenched (almost-sure) volume growth asymptotics for the CRT are deduced, which show that the behaviour in almost-every realisation is not uniform. Applying the results introduced above, these yield heat kernel bounds for the CRT, demonstrating that heat kernel fluctuations occur almost-surely. Finally, a new representation of the CRT as a random self-similar dendrite is presented.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with,the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form. (c) 2007 Elsevier B.V. All rights reserved