5,450 research outputs found
Random recursive trees and the Bolthausen-Sznitman coalescent
We describe a representation of the Bolthausen-Sznitman coalescent in terms
of the cutting of random recursive trees. Using this representation, we prove
results concerning the final collision of the coalescent restricted to [n]: we
show that the distribution of the number of blocks involved in the final
collision converges as n tends to infinity, and obtain a scaling law for the
sizes of these blocks. We also consider the discrete-time Markov chain giving
the number of blocks after each collision of the coalescent restricted to [n];
we show that the transition probabilities of the time-reversal of this Markov
chain have limits as n tends to infinity. These results can be interpreted as
describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in
which a giant cluster containing almost all of the mass has already formed and
the remaining small blocks are being absorbed.Comment: 28 pages, 2 figures. Revised version with minor alterations. To
appear in Electron. J. Proba
Maybe Definitely â Definitely Maybe? EC Competition Law â Is the Time Ripe for Reform?
[Summary]. The aim of this article is to discuss whether the timing for the Commissionâs Proposal for reforming the implementation of Articles 81 and 82 of the EC Treaty is appropriate based on legal certainty considerations. The Proposal suggests to decentralise the day-to-day application of the EC antitrust rules further than is the case today and to abolish the present notification system whereby undertakings can apply for exemptions pursuant to Article 81(3) and negative clearances. The article provides examples showing that presently, the answers to certain legal questions of EC competition rules are vague or contradictory, and that if the proposed reform were to be implemented in its present form, the undertakings would have to carry the full risk for compliance with the competition rules but without a simple or straightforward way of obtaining guidance or legally binding exemptions or negative clearances. While recognising the need for a reform of the implementation system of the EC competition rules, the article argues that consistency and coherence in the understanding and application of the competition rules are a prerequisite to ensure legal certainty which, in turn, is a prerequisite for the implementation of the Commissionâs plans to decentralise EC competition law application. It is therefore suggested that the time is not yet ripe for the type of reform proposed by the Commission and that efforts should instead focus on creating the basis for such a reform by adopting clear guidelines and/or binding legislation that secure the legal certainty of the undertakings that have to operate under the EC competition law framework
Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees
In this paper we give a new example of duality between fragmentation and
coagulation operators. Consider the space of partitions of mass (i.e.,
decreasing sequences of nonnegative real numbers whose sum is 1) and the
two-parameter family of Poisson--Dirichlet distributions that take values in this space. We introduce families of
random fragmentation and coagulation operators and
, respectively, with the following property: if
the input to has
distribution, then the output has
distribution, while the reverse is true for .
This result may be proved using a subordinator representation and it provides a
companion set of relations to those of Pitman between and . Repeated
application of the operators gives rise to a family
of fragmentation chains. We show that these Markov chains can be encoded
naturally by certain random recursive trees, and use this representation to
give an alternative and more concrete proof of the coagulation--fragmentation
duality.Comment: Published at http://dx.doi.org/10.1214/105051606000000655 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Dynamical solutions of a quantum Heisenberg spin glass model
We consider quantum-dynamical phenomena in the ,
infinite-range quantum Heisenberg spin glass. For a fermionic generalization of
the model we formulate generic dynamical self-consistency equations. Using the
Popov-Fedotov trick to eliminate contributions of the non-magnetic fermionic
states we study in particular the isotropic model variant on the spin space.
Two complementary approximation schemes are applied: one restricts the quantum
spin dynamics to a manageable number of Matsubara frequencies while the other
employs an expansion in terms of the dynamical local spin susceptibility. We
accurately determine the critical temperature of the spin glass to
paramagnet transition. We find that the dynamical correlations cause an
increase of by 2% compared to the result obtained in the spin-static
approximation. The specific heat exhibits a pronounced cusp at .
Contradictory to other reports we do not observe a maximum in the -curve
above .Comment: 8 pages, 7 figure
NICMOS Observations of Low-Redshift Quasar Host Galaxies
We have obtained Near-Infrared Camera and Multi-Object Spectrometer images of
16 radio quiet quasars observed as part of a project to investigate the
``luminosity/host-mass limit.'' The limit results were presented in McLeod,
Rieke, & Storrie-Lombardi (1999). In this paper, we present the images
themselves, along with 1- and 2-dimensional analyses of the host galaxy
properties. We find that our model-independent 1D technique is reliable for use
on ground-based data at low redshifts; that many radio-quiet quasars live in
deVaucouleurs-law hosts, although some of the techniques used to determine host
type are questionable; that complex structure is found in many of the hosts,
but that there are some hosts that are very smooth and symmetric; and that the
nuclei radiate at ~2-20% of the Eddington rate based on the assumption that all
galaxies have central black holes with a constant mass fraction of 0.6%.
Despite targeting hard-to-resolve hosts, we have failed to find any that imply
super-Eddington accretion rates.Comment: To appear in ApJ, 28 pages including degraded figures. Download the
paper with full-resolutio figures from
http://www.astro.wellesley.edu/kmcleod/mm.p
Localization of a polymer in random media: Relation to the localization of a quantum particle
In this paper we consider in detail the connection between the problem of a
polymer in a random medium and that of a quantum particle in a random
potential. We are interested in a system of finite volume where the polymer is
known to be {\it localized} inside a low minimum of the potential. We show how
the end-to-end distance of a polymer which is free to move can be obtained from
the density of states of the quantum particle using extreme value statistics.
We give a physical interpretation to the recently discovered one-step
replica-symmetry-breaking solution for the polymer (Phys. Rev. E{\bf 61}, 1729
(2000)) in terms of the statistics of localized tail states. Numerical
solutions of the variational equations for chains of different length are
performed and compared with quenched averages computed directly by using the
eigenfunctions and eigenenergies of the Schr\"odinger equation for a particle
in a one-dimensional random potential. The quantities investigated are the
radius of gyration of a free gaussian chain, its mean square distance from the
origin and the end-to-end distance of a tethered chain. The probability
distribution for the position of the chain is also investigated. The glassiness
of the system is explained and is estimated from the variance of the measured
quantities.Comment: RevTex, 44 pages, 13 figure
A comparison of the optical properties of radio-loud and radio-quiet quasars
We have made radio observations of 87 optically selected quasars at 5 GHz
with the VLA in order to measure the radio power for these objects and hence
determine how the fraction of radio-loud quasars varies with redshift and
optical luminosity. The sample has been selected from the recently completed
Edinburgh Quasar Survey and covers a redshift range of 0.3 < z < 1.5 and an
optical absolute magnitude range of -26.5 < M_{B} < -23.5 (h, q_{0} = 1/2). We
have also matched up other existing surveys with the FIRST and NVSS radio
catalogues and combined these data so that the optical luminosity-redshift
plane is now far better sampled than previously. We have fitted a model to the
probability of a quasar being radio-loud as a function of absolute magnitude
and redshift and from this model infer the radio-loud and radio-quiet optical
luminosity functions. The radio-loud optical luminosity function is featureless
and flatter than the radio-quiet one. It evolves at a marginally slower rate if
quasars evolve by density evolution, but the difference in the rate of
evolutions of the two different classes is much less than was previously
thought. We show, using Monte-Carlo simulations, that the observed difference
in the shape of the optical luminosity functions can be partly accounted for by
Doppler boosting of the optical continuum of the radio-loud quasars and explain
how this can be tested in the future.Comment: 33 pages, 9 postscript figures, uses the AAS aaspp4 LaTeX style file,
to appear in the 1 February 1999 issue of The Astrophysical Journa
Directed polymers on a Cayley tree with spatially correlated disorder
In this paper we consider directed walks on a tree with a fixed branching
ratio K at a finite temperature T. We consider the case where each site (or
link) is assigned a random energy uncorrelated in time, but correlated in the
transverse direction i.e. within the shell. In this paper we take the
transverse distance to be the hierarchical ultrametric distance, but other
possibilities are discussed. We compute the free energy for the case of
quenched disorder and show that there is a fundamental difference between the
case of short range spatial correlations of the disorder which behaves
similarly to the non-correlated case considered previously by Derrida and Spohn
and the case of long range correlations which has a totally different overlap
distribution which approaches a single delta function about q=1 for large L,
where L is the length of the walk. In the latter case the free energy is not
extensive in L for the intermediate and also relevant range of L values,
although in the true thermodynamic limit extensivity is restored. We identify a
crossover temperature which grows with L, and whenever T<T_c(L) the system is
always in the low temperature phase. Thus in the case of long-ranged
correlation as opposed to the short-ranged case a phase transition is absent.Comment: 23 pages, 1 figure, standard latex. J. Phys. A, accepted for
publicatio
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