Structural rigidity concepts are used to understand the origin of instabilities in granular aggregates. It is first demonstrated that the contact network of a noncohesive granular aggregate becomes exactly isostatic when I = kǫ/f l >> 1, where k is stiffness, ǫ is the typical interparticle gap and fL is the typical stress induced by loads. Thus random packings of stiff particles are typically isostatic. Furthermore isostaticity is responsible for the anomalously large susceptibility to perturbation observed in granular aggregates. The load-stress response function of granular piles is critical (power-law distributed) in the isostatic limit, which means that slight overloads will produce internal rearrangements.Photoelastic visualization experiments [1][2][3] show clearly defined stress-concentration paths in non-cohesive granular materials under applied load. These often suffer sudden rearrangement on a global scale when the load conditions are slightly changed, evidencing a degree of susceptibility to perturbation not usually present in elastic materials. It is rather possible that this intrinsic instability be responsible for much of the interesting phenomenology of granular materials [3,4]. Recently a number of phenomenological models [2,5-8] have been put forward, which succeed to reproduce several aspects of stress propagation in granular systems, and the issue of instability has been addressed by noting that the loadstress response function may take negative values [6]. It is the purpose of this letter to show that structural rigidity concepts help us understanding the origin of instability in granular materials, linking it to the topological properties of the system's contact network.Structural rigidity [9] studies the conditions that a network of points connected by rotatable bars (representing central forces) has to fulfill in order to sustain applied loads. A network with too few bars is flexible, while if it has the minimum number required to be rigid it is isostatic. Networks with bars in excess of minimal rigidity are overconstrained, and are in general self-stressed. Concepts from structural rigidity were first introduced in the study of granular media by Guyon et al [10], who stressed that granular systems are not entirely equivalent to linear elastic networks since in the former only compressive interparticle forces are possible. We next show that this constraint has far-reaching consequences for the static behavior of stiff granular aggregates.Consider a d-dimensional frictionless granular pile in equilibrium under the action of external forces F i (gravitational, etc) on its particles. Imagine building an equivalent linear-elastic central-force network (the contact network ), in which two sites are connected by a bond if and only if there is a nonzero compression force between the two corresponding particles. Because of linearity, stresses f ij on the bonds of this equivalent system can be decomposed as f ij = f self ij + f load ij where f self ij are self-stresses, and f load ij are load-depen...

We show that negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heat can be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.

We use a new algorithm to find the stress-carrying backbone of "generic" site-diluted triangular lattices of up to 10 6 sites. Generic lattices can be made by randomly displacing the sites of a regular lattice (see Fig. 1). The percolation threshold is pc = 0.6975±0.0003, the correlation length exponent ν = 1.16 ± 0.03 and the fractal dimension of the backbone D b = 1.78 ± 0.02. The number of "critical bonds" (if you remove them rigidity is lost) on the backbone scales as L x , with x = 0.85 ± 0.05. The Young's modulus is also calculated.

Spreading according to simple rules (e.g., of fire or diseases) and shortest-path distances are studied on d-dimensional systems with a small density p per site of long-range connections ("small-world" lattices). The volume V(t) covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t. From this, the average shortest-path distance l(r) can be calculated as a function of Euclidean distance r. It is found that l(r) approximately r for rr(c). The characteristic length r(c), which governs the behavior of shortest-path lengths, diverges logarithmically with L for all p>0.

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We discuss shortest-path lengths ℓ(r) on periodic rings of size L supplemented with an average of pL randomly located long-range links whose lengths are distributed according to P l ∼ l −µ . Using rescaling arguments and numerical simulation on systems of up to 10 7 sites, we show that a characteristic length ξ exists such that ℓ(r) ∼ r for r < ξ but ℓ(r) ∼ r θs (µ) for r >> ξ. For small p we find that the shortest-path length satisfies the scaling relation ℓ(r, µ, p)/ξ = f (µ, r/ξ). Three regions with different asymptotic behaviors are found, respectively: a) µ > 2 where θs = 1, b) 1 < µ < 2 where 0 < θs(µ) < 1/2 and, c) µ < 1 where ℓ(r) behaves logarithmically, i.e. θs = 0. The characteristic length ξ is of the form ξ ∼ p −ν with ν = 1/(2 − µ) in region b), but depends on L as well in region c). A directed model of shortest-paths is solved and compared with numerical results.

We present an advanced algorithm for the determination of watershed lines on Digital Elevation Models (DEMs), which is based on the iterative application of Invasion Percolation (IIP) . The main advantage of our method over previosly proposed ones is that it has a sub-linear time-complexity. This enables us to process systems comprised of up to 10 8 sites in a few cpu seconds. Using our algorithm we are able to demonstrate, convincingly and with high accuracy, the fractal character of watershed lines.We find the fractal dimension of watersheds to be D f = 1.211 ± 0.001 for artificial landscapes, D f = 1.10 ± 0.01 for the Alpes and D f = 1.11 ± 0.01 for the Himalaya.

Tree models for rigidity percolation are introduced and solved. A probability vector describes the propagation of rigidity outward from a rigid border. All components of this "vector order parameter" are singular at the same rigidity threshold, pc. The infinite-cluster probability P∞ is usually first-order at pc, but often behaves as P∞ ∼ ∆P∞ +(p−pc) 1/2 , indicating critical fluctuations superimposed on a first order jump. Our tree models for rigidity are in qualitative disagreement with "contraint counting" mean field theories. In an important sub-class of tree models "Bootstrap" percolation and rigidity percolation are equivalent.