186 research outputs found
First Passage Properties of the Erdos-Renyi Random Graph
We study the mean time for a random walk to traverse between two arbitrary
sites of the Erdos-Renyi random graph. We develop an effective medium
approximation that predicts that the mean first-passage time between pairs of
nodes, as well as all moments of this first-passage time, are insensitive to
the fraction p of occupied links. This prediction qualitatively agrees with
numerical simulations away from the percolation threshold. Near the percolation
threshold, the statistically meaningful quantity is the mean transit rate,
namely, the inverse of the first-passage time. This rate varies
non-monotonically with p near the percolation transition. Much of this behavior
can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma
Quasi-Exact Solvability and the direct approach to invariant subspaces
We propose a more direct approach to constructing differential operators that
preserve polynomial subspaces than the one based on considering elements of the
enveloping algebra of sl(2). This approach is used here to construct new
exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line
which are not Lie-algebraic. It is also applied to generate potentials with
multiple algebraic sectors. We discuss two illustrative examples of these two
applications: an interesting generalization of the Lam\'e potential which
posses four algebraic sectors, and a quasi-exactly solvable deformation of the
Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure
On large deviation properties of Erdos-Renyi random graphs
We show that large deviation properties of Erd\"os-R\'enyi random graphs can
be derived from the free energy of the -state Potts model of statistical
mechanics. More precisely the Legendre transform of the Potts free energy with
respect to is related to the component generating function of the graph
ensemble. This generalizes the well-known mapping between typical properties of
random graphs and the limit of the Potts free energy. For
exponentially rare graphs we explicitly calculate the number of components, the
size of the giant component, the degree distributions inside and outside the
giant component, and the distribution of small component sizes. We also perform
numerical simulations which are in very good agreement with our analytical
work. Finally we demonstrate how the same results can be derived by studying
the evolution of random graphs under the insertion of new vertices and edges,
without recourse to the thermodynamics of the Potts model.Comment: 38 pages, 9 figures, Latex2e, corrected and extended version
including numerical simulation result
Dynamics of gelling liquids: a short survey
The dynamics of randomly crosslinked liquids is addressed via a Rouse- and a
Zimm-type model with crosslink statistics taken either from bond percolation or
Erdoes-Renyi random graphs. While the Rouse-type model isolates the effects of
the random connectivity on the dynamics of molecular clusters, the Zimm-type
model also accounts for hydrodynamic interactions on a preaveraged level. The
incoherent intermediate scattering function is computed in thermal equilibrium,
its critical behaviour near the sol-gel transition is analysed and related to
the scaling of cluster diffusion constants at the critical point. Second,
non-equilibrium dynamics is studied by looking at stress relaxation in a simple
shear flow. Anomalous stress relaxation and critical rheological properties are
derived. Some of the results contradict long-standing scaling arguments, which
are shown to be flawed by inconsistencies.Comment: 21 pages, 3 figures; Dedicated to Lothar Schaefer on the occasion of
his 60th birthday; Changes: added comments on the gel phase and some
reference
First Light Measurements with the XMM-Newton Reflection Grating Spectrometers: Evidence for an Inverse First Ionisation Potential Effect and Anomalous Ne A bundance in the Coronae of HR 1099
The RS CVn binary system HR 1099 was extensively observed by the XMM-Newton
observatory in February 2000 as its first-light target. A total of 570 ks of
exposure time was accumulated with the Reflection Grating Spectrometers (RGS).
The integrated X-ray spectrum between 5-35 Angstrom is of unprecedented quality
and shows numerous features attributed to transitions of the elements C, N, O,
Ne, Mg, Si, S, Fe, and Ni. We perform an in-depth study of the elemental
composition of the average corona of this system, and find that the elemental
abundances strongly depend on the first ionisation potential (FIP) of the
elements. But different from the solar coronal case, we find an inverse FIP
effect, i.e., the abundances (relative to oxygen) increase with increasing FIP.
Possible scenarios, e.g., selective enrichment due to Ne-rich flare-like
events, are discussed.Comment: 5 pages, 4 figures. Accepted by A&A Letters, XMM issu
A Mission to Explore the Pioneer Anomaly
The Pioneer 10 and 11 spacecraft yielded the most precise navigation in deep
space to date. These spacecraft had exceptional acceleration sensitivity.
However, analysis of their radio-metric tracking data has consistently
indicated that at heliocentric distances of astronomical units,
the orbit determinations indicated the presence of a small, anomalous, Doppler
frequency drift. The drift is a blue-shift, uniformly changing with a rate of
Hz/s, which can be interpreted as a
constant sunward acceleration of each particular spacecraft of . This signal has become known as the Pioneer
anomaly. The inability to explain the anomalous behavior of the Pioneers with
conventional physics has contributed to growing discussion about its origin.
There is now an increasing number of proposals that attempt to explain the
anomaly outside conventional physics. This progress emphasizes the need for a
new experiment to explore the detected signal. Furthermore, the recent
extensive efforts led to the conclusion that only a dedicated experiment could
ultimately determine the nature of the found signal. We discuss the Pioneer
anomaly and present the next steps towards an understanding of its origin. We
specifically focus on the development of a mission to explore the Pioneer
Anomaly in a dedicated experiment conducted in deep space.Comment: 8 pages, 9 figures; invited talk given at the 2005 ESLAB Symposium
"Trends in Space Science and Cosmic Vision 2020", 19-21 April 2005, ESTEC,
Noordwijk, The Netherland
Fundamental Physics with the Laser Astrometric Test Of Relativity
The Laser Astrometric Test Of Relativity (LATOR) is a joint European-U.S.
Michelson-Morley-type experiment designed to test the pure tensor metric nature
of gravitation - a fundamental postulate of Einstein's theory of general
relativity. By using a combination of independent time-series of highly
accurate gravitational deflection of light in the immediate proximity to the
Sun, along with measurements of the Shapiro time delay on interplanetary scales
(to a precision respectively better than 0.1 picoradians and 1 cm), LATOR will
significantly improve our knowledge of relativistic gravity. The primary
mission objective is to i) measure the key post-Newtonian Eddington parameter
\gamma with accuracy of a part in 10^9. (1-\gamma) is a direct measure for
presence of a new interaction in gravitational theory, and, in its search,
LATOR goes a factor 30,000 beyond the present best result, Cassini's 2003 test.
The mission will also provide: ii) first measurement of gravity's non-linear
effects on light to ~0.01% accuracy; including both the Eddington \beta
parameter and also the spatial metric's 2nd order potential contribution (never
measured before); iii) direct measurement of the solar quadrupole moment J2
(currently unavailable) to accuracy of a part in 200 of its expected size; iv)
direct measurement of the "frame-dragging" effect on light by the Sun's
gravitomagnetic field, to 1% accuracy. LATOR's primary measurement pushes to
unprecedented accuracy the search for cosmologically relevant scalar-tensor
theories of gravity by looking for a remnant scalar field in today's solar
system. We discuss the mission design of this proposed experiment.Comment: 8 pages, 9 figures; invited talk given at the 2005 ESLAB Symposium
"Trends in Space Science and Cosmic Vision 2020," 19-21 April 2005, ESTEC,
Noodrwijk, The Netherland
The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma
We elucidate the close connection between the repulsive lattice gas in
equilibrium statistical mechanics and the Lovasz local lemma in probabilistic
combinatorics. We show that the conclusion of the Lovasz local lemma holds for
dependency graph G and probabilities {p_x} if and only if the independent-set
polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore,
we show that the usual proof of the Lovasz local lemma -- which provides a
sufficient condition for this to occur -- corresponds to a simple inductive
argument for the nonvanishing of the independent-set polynomial in a polydisc,
which was discovered implicitly by Shearer and explicitly by Dobrushin. We also
present some refinements and extensions of both arguments, including a
generalization of the Lovasz local lemma that allows for "soft" dependencies.
In addition, we prove some general properties of the partition function of a
repulsive lattice gas, most of which are consequences of the alternating-sign
property for the Mayer coefficients. We conclude with a brief discussion of the
repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity.
To be published in J. Stat. Phy
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