Analogue Models of and for Gravity

Condensed matter systems, such as acoustics in flowing fluids, light in moving dielectrics, or quasiparticles in a moving superfluid, can be used to mimic aspects of general relativity. More precisely these systems (and others) provide experimentally accessible models of curved-space quantum field theory. As such they mimic kinematic aspects of general relativity, though typically they do not mimic the dynamics. Although these analogue models are thereby limited in their ability to duplicate all the effects of Einstein gravity they nevertheless are extremely important—they provide black hole analogues (some of which have already been seen experimentally) and lead to tests of basic principles of curved-space quantum field theory. Currently these tests are still in the realm of gedanken-experiments, but there are plausible candidate models that should lead to laboratory experiments in the not too distant future.     

Spatial design matrices and associated quadratic forms: structure and properties

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.     

A stable static universe?

Starting from the assumption that general relativity might be an emergent phenomenon showing up at low energies from an underlying microscopic structure, we reanalyze the stability of a static closed universe filled with radiation. In this scenario, it is sensible to consider the effective general-relativistic configuration as in a thermal contact with an “environment” (the role of the environment can be played, for example, by a higher-dimensional bulk or by the trans-Planckian degrees of freedom). We calculate the free energy at a fixed temperature of this radiation-filled static configuration. Then, by looking at the free energy, we show that the static Einstein configuration is stable under the stated condition.     

Mapping Properties of a Projection Related to the Helmholtz Equation

We study the continuity properties of a projection derived from a recent characterization of Herglotz Wave Functions in the plane. Herglotz Wave Functions are the entire solutions of the Helmholtz equation which have L²-Far-Field-Pattern. The behavior of this projection is reminiscent of the Disc Multiplier Operator on both L^p and mixed L^p-normed spaces. In particular a Kakeya type set plays a central role in our analysis. As a consequence we answer a question posed by Alvarez, Folch-Gabayet and Pérez-Esteva.     

Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

Flow through a porous medium with mass removal and diffusion

A one-dimensional model of filtration through a saturated porous medium with a mutual action between the solid matrix and the flow is investigated. The substances removed from the porous soil are both particles transported by the liquid flow and substances which diffuse in the fluid. Received June 30, 1997     

The effect of the anion on the physical properties of trihalide-based N,N-dialkylimidazolium ionic liquids

Low viscosity, high density trihalide-based 1-n-butyl-3-methylimidazolium ionic liquids have been prepared and characterised. Key physical properties (density, conductivity, melting point, refractive index, surface tension and diffusion coefficient) of the ionic liquids have been determined and are compared with those of other 1,3-dialkylimidazolium molten salts. The relationship between anion identity and the physical properties of the ionic liquids under investigation is discussed.