Moments in graphs

Let $G$ be a connected graph with vertex set $V$ and a {\em weight function} $\rho$ that assigns a nonnegative number to each of its vertices. Then, the {\em $\rho$-moment} of $G$ at vertex $u$ is defined to be M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot) stands for the distance function. Adding up all these numbers, we obtain the {\em $\rho$-moment of $G$}: M_G^{\rho}=\sum_{u\in V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\em Wiener index} $W(G)$, when $\rho(u)=1/2$ for every $u\in V$, and the {\em degree distance} $D'(G)$, obtained when $\rho(u)=\delta(u)$, the degree of vertex $u$. In this paper we derive some exact formulas for computing the $\rho$-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding $\rho$-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same $\rho$-moment for every $\rho$ (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product

Bubble fluctuations in Ω<1\Omega<1 inflation

In the context of the open inflationary universe, we calculate the amplitude of quantum fluctuations which deform the bubble shape. These give rise to scalar field fluctuations in the open Friedman-Robertson-Walker universe which is contained inside the bubble. One can transform to a new gauge in which matter looks perfectly smooth, and then the perturbations behave as tensor modes (gravitational waves of very long wavelength). For $(1-\Omega)<<1$, where $\Omega$ is the density parameter, the microwave temperature anisotropies produced by these modes are of order $\delta T/T\sim H(R_0\mu l)^{-1/2} (1-\Omega)^{l/2}$. Here, $H$ is the expansion rate during inflation, $R_0$ is the intrinsic radius of the bubble at the time of nucleation, $\mu$ is the bubble wall tension and $l$ labels the different multipoles ($l>1$). The gravitational backreaction of the bubble has been ignored. In this approximation, $G\mu R_0<<1$, and the new effect can be much larger than the one due to ordinary gravitational waves generated during inflation (unless, of course, $\Omega$ gets too close to one, in which case the new effect disappears).Comment: 17 pages, 3 figs, LaTeX, epsfig.sty, available at ftp://ftp.ifae.es/preprint/ft/uabft387.p

Zero-th law in structural glasses: an example

We investigate the validity of a zeroth thermodynamic law for non-equilibrium systems. In order to describe the thermodynamics of the glassy systems, it has been introduced an extra parameter, the effective temperature which generalizes the fluctuation-dissipation theorem (FDT) to off-equilibrium systems and supposedly describes thermal fluctuations around the aging state. In particular we analyze two coupled systems of harmonic oscillators with Monte Carlo dynamics. We study in detail two types of dynamics: sequential dynamics, where the coupling between the subsystems comes only from the Hamiltonian; and parallel dynamics where there is another source of coupling: the dynamics. We show how in the first case the effective temperatures of the two interacting subsystems are different asymptotically due to the smallness of the thermal conductivity in the aging regime. This explains why, in structural glasses, different interacting degrees of freedom can stay at different effective temperatures, and never thermalize.Comment: 10 pages. Contribution to the Proceedings of the ESF SPHINX meeting `Glassy behaviour of kinetically constrained models' (Barcelona, March 22-25, 2001). To appear in a special issue of J. Phys. Cond. Mat

Gravity in the Randall-Sundrum Brane World

We discuss the weak gravitational field created by isolated matter sources in the Randall-Sundrum brane-world. In the case of two branes of opposite tension, linearized Brans-Dicke (BD) gravity is recovered on either wall, with different BD parameters. On the wall with positive tension the BD parameter is larger than 3000 provided that the separation between walls is larger than 4 times the AdS radius. For the wall of negative tension, the BD parameter is always negative but greater than -3/2. In either case, shadow matter from the other wall gravitates upon us. For equal Newtonian mass, light deflection from shadow matter is 25 % weaker than from ordinary matter. Hence, the effective mass of a clustered object containing shadow dark matter would be underestimated if naively measured through its lensing effect. For the case of a single wall of positive tension, Einstein gravity is recovered on the wall to leading order, and if the source is stationary the field stays localized near the wall. We calculate the leading Kaluza-Klein corrections to the linearized gravitational field of a non-relativistic spherical object and find that the metric is different from the Schwarzschild solution at large distances. We believe that our linearized solution corresponds to the field far from the horizon after gravitational collapse of matter on the brane.Comment: 8 pages, 1 figure. Replaced with revised version to be published in Phys. Rev. Lett. Some comments adde

Second Order Perturbations of a Macroscopic String; Covariant Approach

Using a world-sheet covariant formalism, we derive the equations of motion for second order perturbations of a generic macroscopic string, thus generalizing previous results for first order perturbations. We give the explicit results for the first and second order perturbations of a contracting near-circular string; these results are relevant for the understanding of the possible outcome when a cosmic string contracts under its own tension, as discussed in a series of papers by Vilenkin and Garriga. In particular, second order perturbations are necessaary for a consistent computation of the energy. We also quantize the perturbations and derive the mass-formula up to second order in perturbations for an observer using world-sheet time $\tau$. The high frequency modes give the standard Minkowski result while, interestingly enough, the Hamiltonian turns out to be non-diagonal in oscillators for low-frequency modes. Using an alternative definition of the vacuum, it is possible to diagonalize the Hamiltonian, and the standard string mass-spectrum appears for all frequencies. We finally discuss how our results are also relevant for the problems concerning string-spreading near a black hole horizon, as originally discussed by Susskind.Comment: New discussion about the quantum mass-spectrum in chapter

Gravity Waves from Instantons

We perform a first principles computation of the spectrum of gravity waves produced in open inflationary universes. The background spacetime is taken to be the continuation of an instanton saddle point of the Euclidean no boundary path integral. The two-point tensor correlator is computed directly from the path integral and is shown to be unique and well behaved in the infrared. We discuss the tensor contribution to the cosmic microwave background anisotropy and show how it may provide an observational discriminant between different types of primordial instantons.Comment: 19 pages, RevTex file, including two postscript figure file