Field Theory And Second Renormalization Group For Multifractals In Percolation

The field-theory for multifractals in percolation is reformulated in such a way that multifractal exponents clearly appear as eigenvalues of a second renormalization group. The first renormalization group describes geometrical properties of percolation clusters, while the second-one describes electrical properties, including noise cumulants. In this context, multifractal exponents are associated with symmetry-breaking fields in replica space. This provides an explanation for their observability. It is suggested that multifractal exponents are ''dominant'' instead of ''relevant'' since there exists an arbitrary scale factor which can change their sign from positive to negative without changing the Physics of the problem.Comment: RevTex, 10 page

Topological Graph Inverse Semigroups

To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$. On the other hand, $G(E)$ need not be discrete in a Hausdorff semigroup topology, and for certain graphs $E$, $G(E)$ admits a $T_1$ semigroup topology in which $G(E)\setminus \{0\}$ is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of $G(E)$ in larger topological semigroups.Peer reviewe

Geometric origin of mechanical properties of granular materials

Some remarkable generic properties, related to isostaticity and potential energy minimization, of equilibrium configurations of assemblies of rigid, frictionless grains are studied. Isostaticity -the uniqueness of the forces, once the list of contacts is known- is established in a quite general context, and the important distinction between isostatic problems under given external loads and isostatic (rigid) structures is presented. Complete rigidity is only guaranteed, on stability grounds, in the case of spherical cohesionless grains. Otherwise, the network of contacts might deform elastically in response to load increments, even though grains are rigid. This sets an uuper bound on the contact coordination number. The approximation of small displacements (ASD) allows to draw analogies with other model systems studied in statistical mechanics, such as minimum paths on a lattice. It also entails the uniqueness of the equilibrium state (the list of contacts itself is geometrically determined) for cohesionless grains, and thus the absence of plastic dissipation. Plasticity and hysteresis are due to the lack of such uniqueness and may stem, apart from intergranular friction, from small, but finite, rearrangements, in which the system jumps between two distinct potential energy minima, or from bounded tensile contact forces. The response to load increments is discussed. On the basis of past numerical studies, we argue that, if the ASD is valid, the macroscopic displacement field is the solution to an elliptic boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and minor errors correcte