Combinatorial models of rigidity and renormalization

We first introduce the percolation problems associated with the graph theoretical concepts of $(k,l)$-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for $(k,l)$-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of $(k,l)$-percolation problems on hierarchical graphs, for $k\leq l<2k$. We introduce and solve by renormalization such a model, which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions, depending on the values of the parameters.Comment: 22 pages, 6 figure

Percolation in an ultrametric space

We study percolation on the hierarchical lattice of order $N$ where the probability of connection between two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)},\; \delta >-1$. Since the distance is an ultrametric, there are significant differences with percolation on the Euclidean lattice. There are two non-critical regimes: $\delta <1$, where percolation occurs, and $\delta >1$, where it does not occur. In the critical case, $\delta =1$, we use an approach in the spirit of the renormalization group method of statistical physics and connectivity results of Erd\H{o}s-Renyi random graphs play a key role. We find sufficient conditions on $c_k$ such that percolation occurs, or that it does not occur. An intermediate situation called pre-percolation is also considered. In the cases of percolation we prove uniqueness of the constructed percolation clusters. In a previous paper \cite{DG1} we studied percolation in the $N\to\infty$ limit (mean field percolation) which provided a simplification that allowed finding a necessary and sufficient condition for percolation. For fixed $N$ there are open questions, in particular regarding the existence of a critical value of a parameter in the definition of $c_k$, and if it exists, what would be the behaviour at the critical point

Conducting-angle-based percolation in the XY model

We define a percolation problem on the basis of spin configurations of the two dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.Comment: 16 pages, 14 figure