Critical random hypergraphs: The emergence of a giant set of identifiable vertices

We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase transition takes various forms, depending on the values of the parameters controlling the different types of hyperedges. It may be continuous as in a random graph. (In fact, when there are no higher-order edges, it is exactly the emergence of the giant component.) In this case, there is a sequence of possible sizes of ``components'' (including but not restricted to N^{2/3}). Alternatively, the phase transition may be discontinuous. We are particularly interested in the nature of the discontinuous phase transition and are able to exhibit precise asymptotics. Our method extends a result of Aldous [Ann. Probab. 25 (1997) 812-854] on component sizes in a random graph.Comment: Published at http://dx.doi.org/10.1214/009117904000000847 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Manifolds in random media: A variational approach to the spatial probability distribution

We develop a new variational scheme to approximate the position dependent spatial probability distribution of a zero dimensional manifold in a random medium. This celebrated 'toy-model' is associated via a mapping with directed polymers in 1+1 dimension, and also describes features of the commensurate-incommensurate phase transition. It consists of a pointlike 'interface' in one dimension subject to a combination of a harmonic potential plus a random potential with long range spatial correlations. The variational approach we develop gives far better results for the tail of the spatial distribution than the hamiltonian version, developed by Mezard and Parisi, as compared with numerical simulations for a range of temperatures. This is because the variational parameters are determined as functions of position. The replica method is utilized, and solutions for the variational parameters are presented. In this paper we limit ourselves to the replica symmetric solution.Comment: 22 pages, 3 figures available on request, Revte

Alfred Müller-Armack and Ludwig Erhard: Social Market Liberalism

"Soziale Marktwirtschaft" (Social Market Economy) is the economic order that was established in Western Germany after 1945. It is not a precisely outlined theoretical system but more a cipher for a "mélange" of socio-political ideas for a free and socially just society and some general rules of economic policy. It is a decided liberal concept, based on individual freedom and the belief that well-functioning markets and competition lead to economic efficiency and by this, to economic development (or in the case of Germany, recovery) and social improvement. But in sharp distinction to the harmonious Smithian world of the "invisible hand", the "founding fathers" of the post-war economic order in Germany were convinced that the economic system must be guided by an "economic constitution" provided by the state. --

The Brownian continuum random tree as the unique solution to a fixed point equation

In this note, we provide a new characterization of Aldous' Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.Comment: 15 pages, 3 figure

Essential edges in Poisson random hypergraphs

Consider a random hypergraph on a set of N vertices in which, for k between 1 and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all subsets of size k. We collapse the hypergraph by running the following algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse the hyperedges over that vertex onto their remaining vertices; repeat until there are no 1-edges left. We call the vertices removed in this process "identifiable". Also any hyperedge all of whose vertices are removed is called "identifiable". We say that a hyperedge is "essential" if its removal prior to collapse would have reduced the number of identifiable vertices. The limiting proportions, as N tends to infinity, of identifiable vertices and hyperedges were obtained by Darling and Norris. In this paper, we establish the limiting proportion of essential hyperedges. We also discuss, in the case of a random graph, the relation of essential edges to the 2-core of the graph, the maximal sub-graph with minimal vertex degree 2.Comment: 12 pages, 3 figures. Revised version with minor corrections/clarifications and slightly expanded introductio

Preservation of log-concavity on summation

We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of independent (not necessarily log-concave) random variables

Zero-Energy Fields on Complex Projective Space

We consider complex projective space with its Fubini-Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.Comment: 30 page

Zero-energy fields on complex projective space

We consider complex projective space with its Fubini-Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields