Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Using survey participants to estimate the impact of nonparticipation

The authors evaluate the effectiveness of two models often used to measure the extent of nonparticipation bias in survey estimates. The first model establishes a "continuum of resistance" to being surveyed, placing people who were interviewed after one phone call on one end and nonparticipants on the other. The second assumes that there are "classes" of nonparticipants and that similar classes can be found among participants; it identifies groups of participants thought to be like nonparticipants and uses them as "proxies" to estimate the characteristics of nonparticipants. The authors use these models to examine how accurately they estimate the characteristics of nonparticipants and the impact of nonparticipation on survey estimates of means of child support awards and payments in Wisconsin. They find that neither model detects the true extent of nonparticipation bias.

The Fundamental Plane of Galaxy Clusters

Velocity dispersion $\sigma$, radius $R$ and luminosity $L$ of elliptical galaxies are known to be related, leaving only two degrees of freedom and defining the so-called ``fundamental plane". In this {\em Letter} we present observational evidence that rich galaxy clusters exhibit a similar behaviour. Assuming a relation $L \propto R^{\alpha}\sigma^{2 \beta}$, the best-fit values of $\alpha$ and $\beta$ are very close to those defined by galaxies. The dispersion of this relation is lower than 10 percent, i.e. significantly smaller than the dispersion observed in the $L-\sigma$ and $L-R$ relations. We briefly suggest some possible implications on the spread of formation times of objects and on peculiar velocities of galaxy clusters.Comment: 11pp., 4 figures (available on request), LaTeX, BAP-04-1993-015-OA

Scaling in Gravitational Clustering, 2D and 3D Dynamics

Perturbation Theory (PT) applied to a cosmological density field with Gaussian initial fluctuations suggests a specific hierarchy for the correlation functions when the variance is small. In particular quantitative predictions have been made for the moments and the shape of the one-point probability distribution function (PDF) of the top-hat smoothed density. In this paper we perform a series of systematic checks of these predictions against N-body computations both in 2D and 3D with a wide range of featureless power spectra. In agreement with previous studies, we found that the reconstructed PDF-s work remarkably well down to very low probabilities, even when the variance approaches unity. Our results for 2D reproduce the features for the 3D dynamics. In particular we found that the PT predictions are more accurate for spectra with less power on small scales. The nonlinear regime has been explored with various tools, PDF-s, moments and Void Probability Function (VPF). These studies have been done with unprecedented dynamical range, especially for the 2D case, allowing in particular more robust determinations of the asymptotic behaviour of the VPF. We have also introduced a new method to determine the moments based on the factorial moments. Results using this method and taking into account the finite volume effects are presented.Comment: 13 pages, Latex file, 9 Postscript Figure

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small correction

A Count Probability Cookbook: Spurious Effects and the Scaling Model

We study the errors brought by finite volume effects and dilution effects on the practical determination of the count probability distribution function P_N(n,L), which is the probability of having N objects in a cell of volume L^3 for a set of average number density n. Dilution effects are particularly relevant to the so-called sparse sampling strategy. This work is mainly done in the framework of the scaling model (Balian \& Schaeffer 1989), which assumes that the Q-body correlation functions obey the scaling relation xi_Q(K r_1,..., K r_Q) = K^{-(Q-1) gamma} xi_N(r_1,..., r_Q). We use three synthetic samples as references to perform our analysis: a fractal generated by a Rayleigh-L\'evy random walk with 3.10^4 objects, a sample dominated by a spherical power-law cluster with 3.10^4 objects and a cold dark matter (CDM) universe involving 3.10^5 matter particles.Comment: 44 pages, uuencoded compressed postcript file, FERMILAB-Pub-94/229-A, accepted in ApJ

MACOC: a medoid-based ACO clustering algorithm

The application of ACO-based algorithms in data mining is growing over the last few years and several supervised and unsupervised learning algorithms have been developed using this bio-inspired approach. Most recent works concerning unsupervised learning have been focused on clustering, showing great potential of ACO-based techniques. This work presents an ACO-based clustering algorithm inspired by the ACO Clustering (ACOC) algorithm. The proposed approach restructures ACOC from a centroid-based technique to a medoid-based technique, where the properties of the search space are not necessarily known. Instead, it only relies on the information about the distances amongst data. The new algorithm, called MACOC, has been compared against well-known algorithms (K-means and Partition Around Medoids) and with ACOC. The experiments measure the accuracy of the algorithm for both synthetic datasets and real-world datasets extracted from the UCI Machine Learning Repository

Distance statistics in large toroidal maps

We compute a number of distance-dependent universal scaling functions characterizing the distance statistics of large maps of genus one. In particular, we obtain explicitly the probability distribution for the length of the shortest non-contractible loop passing via a random point in the map, and that for the distance between two random points. Our results are derived in the context of bipartite toroidal quadrangulations, using their coding by well-labeled 1-trees, which are maps of genus one with a single face and appropriate integer vertex labels. Within this framework, the distributions above are simply obtained as scaling limits of appropriate generating functions for well-labeled 1-trees, all expressible in terms of a small number of basic scaling functions for well-labeled plane trees.Comment: 24 pages, 9 figures, minor corrections, new added reference