The relationship between the Jacobi and the successive overrelaxation (SOR) matrices of a k-cyclic matrix

AbstractLet A be a (k−l, l)-generalized consistently ordered matrix with T and Lω its associated Jacobi and SOR matrices whose eigenvalues μ and λ satisfy the well-known relationship (λ+ω−1)k=ωkμkλk−1. For a subclass of the above matrices A we prove that the matrix analogue of the previous relationship holds. Exploiting the matrix relationship we show that the SOR method is equivalent to a certain monoparametric k-step iterative one when used for the solution of the fixed-point problem x=Tx+c

Long-lived Giant Number Fluctuations in a Swarming Granular Nematic

Coherently moving flocks of birds, beasts or bacteria are examples of living matter with spontaneous orientational order. How do these systems differ from thermal equilibrium systems with such liquid-crystalline order? Working with a fluidized monolayer of macroscopic rods in the nematic liquid crystalline phase, we find giant number fluctuations consistent with a standard deviation growing linearly with the mean, in contrast to any situation where the Central Limit Theorem applies. These fluctuations are long-lived, decaying only as a logarithmic function of time. This shows that flocking, coherent motion and large-scale inhomogeneity can appear in a system in which particles do not communicate except by contact.Comment: This is the author's version of the work. It is posted here by permission of the AAAS. The definitive version is to appear in SCIENC

Spatial Mixing and Non-local Markov chains

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra

Lee-yang zeros and the complexity of the ferromagnetic ising model on bounded-degree graphs

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model in the Lee-Yang circle of zeros given by |λ| = 1, where λ is the external field of the model. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all |λ| ≠ 1 by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens in the circular arc around λ = 1, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point λ = 1; in fact, our techniques apply more generally to the whole unit circle |λ| = 1. We show #P-hardness for approximating the partition function on graphs of maximum degree Δ when b, the edge-interaction parameter, is in the interval [EQUATION] and λ is a non-real on the unit circle. This result contrasts with known approximation algorithms when |λ| ≠ 1 or [EQUATION], and shows that the Lee-Yang circle of zeros is computationally intractable, even on bounded-degree graphs. Our inapproximability result is based on constructing rooted tree gadgets via a detailed understanding of the underlying dynamical systems, which are further parameterised by the degree of the root. The ferromagnetic Ising model has radically different behaviour than previously considered anti-ferromagnetic models, and showing our #P-hardness results in the whole Lee-Yang circle requires a new high-level strategy to construct the gadgets. To this end, we devise an elaborate inductive procedure to construct the required gadgets by taking into account the dependence between the degree of the root of the tree and the magnitude of the derivative at the fixpoint of the corresponding dynamical system

Nonequilibrium steady states in a vibrated-rod monolayer: tetratic, nematic and smectic correlations

We study experimentally the nonequilibrium phase behaviour of a horizontal monolayer of macroscopic rods. The motion of the rods in two dimensions is driven by vibrations in the vertical direction. Aside from the control variables of packing fraction and aspect ratio that are typically explored in molecular liquid crystalline systems, due to the macroscopic size of the particles we are also able to investigate the effect of the precise shape of the particle on the steady states of this driven system. We find that the shape plays an important role in determining the nature of the orientational ordering at high packing fraction. Cylindrical particles show substantial tetratic correlations over a range of aspect ratios where spherocylinders have previously been shown by Bates et al (JCP 112, 10034 (2000)) to undergo transitions between isotropic and nematic phases. Particles that are thinner at the ends (rolling pins or bails) show nematic ordering over the same range of aspect ratios, with a well-established nematic phase at large aspect ratio and a defect-ridden nematic state with large-scale swirling motion at small aspect ratios. Finally, long-grain, basmati rice, whose geometry is intermediate between the two shapes above, shows phases with strong indications of smectic order.Comment: 18 pages and 13 eps figures, references adde