We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immediately yield a Maxwell-Laman-type combinatorial characterization for frameworks embedded in 2-dimensional cones that arise as quotients of the plane by a finite order rotation.
The general area of rigidity with symmetry has been somewhat active in the past few years, but the results here are independent of much of it. For completeness, we review some work along similar lines. A specialization of our [Theorem A] MT10 is due to Ross @cite_2 @cite_18 . Schulze @cite_13 @cite_11 and Schulze and Whiteley @cite_16 studied the question of when incidental'' symmetry induces non-generic behaviors in finite frameworks, which is a different setting than the forced'' symmetry we consider here and in @cite_15 . Ross, Schulze, and Whiteley @cite_21 have studied the present problems, but they do not give any combinatorial characterizations. Borcea and Streinu @cite_24 have proposed a kind of doubly generic'' periodic rigidity, where the combinatorial model does not include the colors on the quotient graph.
@cite_18 @cite_21 @cite_24 @cite_2 @cite_15 @cite_16 @cite_13 @cite_11
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Abstract We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks . The characterization is a true analogue of the Maxwell–Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z 2 - graded-sparse colored graphs . For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which types of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem--e.g., dynamical, rigidity, and satisfiability-unsatisfiability (SAT-UNSAT) transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance, we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so-called rigidity transition where the thermodynamically relevant clusters become frozen. These nonequilibrium solutions belong to the most probable unfrozen clusters. The following question is due to Chatterjee and Varadhan 2011. Fix 0<p<r<1 and take G Gn,p, the Erdi¾?s-Renyi random graph with edge density p, conditioned to have at least as many triangles as the typical Gn,r. Is G close in cut-distance to a typical Gn,r? Via a beautiful new framework for large deviation principles in Gn,p, Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of p,r where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d-regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all p,r such that rd,hpr lies on the convex minorant of xi¾?hpx1 d where hp is the rate function of a binomial with parameter p. In particular, the answer for triangles involves hpx rather than the natural guess of hpx1 3 where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of G Gn,p is conditioned to exceed the typical value of the largest eigenvalue of Gn,r. Building on the work of Chatterjee and Diaconis 2012 we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn 2001 and Galvin and Tet ali 2004. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109-146, 2015 We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree T, and study the existence of the free energy density ϕ, the limit of the log-partition function divided by the number of vertices n as n tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ subject to uniqueness of a relevant Gibbs measure for the factor model on T. By way of example we compute ϕ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on T. In the special case that T has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest. A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry. We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface @math in a toric variety @math we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of @math along @math , under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to @math . The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.