Critical line of an n-component cubic model

We consider a special case of the n-component cubic model on the square lattice, for which an expansion exists in Ising-like graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determine the critical points for several values of n. Furthermore we determine several universal quantities, including three critical exponents. For n<2, these results agree well with the theoretical predictions for the critical O(n) branch. This model is also a special case of the ($N_\alpha,N_\beta$) model of Domany and Riedel. It appears that the self-dual plane of the latter model contains the exactly known critical points of the n=1 and 2 cubic models. For this reason we have checked whether this is also the case for 1<n<2. However, this possibility is excluded by our numerical results

Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity $c\simeq 1.43$. The replica-symmetric solution breaks down at $c=e\simeq 2.72$. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.Comment: 32 pages, 9 eps figures, to app. in PRE (01 May 2001

Spectra of Sparse Random Matrices

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.Comment: 22 papges, 8 figures (one on graph-Laplacians added), one reference added, some typos eliminate

Minimizing Unsatisfaction in Colourful Neighbourhoods

Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods, given a predetermined number of colours. In the analytical framework of a tree approximation, carried out at both zero and finite temperatures, solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition, which are consistent with those of existing algorithms. The nature of the transition as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional explanatio

Towards finite-dimensional gelation

We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.Comment: 18 pages, 4 figures, version to be published in EPJ