Bulletin of the American Physical Society
2007 APS March Meeting
Volume 52, Number 1
Monday–Friday, March 5–9, 2007; Denver, Colorado
Session S7: Percolation |
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Sponsoring Units: GSNP Chair: Robert Ziff, University of Michigan Room: Colorado Convention Center Korbel 4A-4B |
Wednesday, March 7, 2007 2:30PM - 3:06PM |
S7.00001: Percolation, Schramm-Loewner evolutions, and applications Invited Speaker: The area of two-dimensional (2D) critical phenomena has enjoyed a recent breakthrough. A radically new development (recognized just a few months ago by a Fields medal) termed the Schramm- (or stochastic) Loewner evolution (SLE), has given new tools to study criticality and conformal invariance in 2D. Percolation was a natural common ground for physicists and mathematicians, and played a crucial role in motivating and shaping up the emergence of SLE as a theoretical and computational tool. The new description focuses directly on non-local structures that characterize a given system, be it a boundary of an Ising or percolation cluster, or loops in the O$(n)$ model. This description uses the fact that all these non-local objects become random curves at a critical point, and may be precisely characterized by stochastic dynamics of certain conformal maps. In my talk I will review this recent development in relation to percolation, as well as touch upon its applications to other areas of physics. [Preview Abstract] |
Wednesday, March 7, 2007 3:06PM - 3:42PM |
S7.00002: 2d Turbulence, percolation and SLE Invited Speaker: We analyze isolines of scalar fields (vorticity, temperature) in different cases of 2d turbulence and found that they belong to the SLE class, i.e. to curves that can be mapped to 1d Brownian motion. Such curves have conformal invariant statistics. We find that vorticity isolines in 2d turbulence are equivalent (within our 5{\%} accuracy) to $SLE_6 $ i.e. to percolation despite the fact that the vorticity field is long-correlated and does not satisfy Harris criterium. We find that the temperature isolines in surface quasi-geostrophic turbulence belong to $SLE_4 $i.e. statistically equivalent to isolines of a Gaussian free field despite the fact that the temperature is non-Gaussian. Link with SLE allows one to obtain a variety of quantitative results going well beyond all we knew about turbulence before and hints about some deep analogy between turbulence and critical phenomena. [Preview Abstract] |
Wednesday, March 7, 2007 3:42PM - 4:18PM |
S7.00003: Percolation properties of complex networks with weak and strong clustering Invited Speaker: A diversity of systems in the real world can be analyzed as complex networks. This makes any theoretical development in the field potentially applicable to many different areas. As a germane example, percolation has helped us to understand, for instance, the high resilience of scale-free networks in front of the random removal of a fraction of their constituents, with important implications for communication or biological systems among others. In addition to its high theoretical interest, it serves as a conceptual approach to treat more factual problems on networks, such as the dynamics of epidemic spreading. On the other hand, when large systems of interactions are mapped into comprehensible graphs, just vertices and edges are usually recognized as the primary building blocks. However, transitive relations, represented by triangles and referred to as clustering, should also be taken into account as a basic structure whose presence and self-organization can drastically impact network structure and properties. In this framework, the introduction of clustering in the percolation analysis of complex networks represents a theoretical challenge. Previous approaches were based on the idea of branching process, which works well when the network is locally treelike and thus the clustering coefficient is very small. Real networks, however, are shown to have a significant level of clustering. They can be classified in networks with weak transitivity, in which triangles are disjoint, and networks with strong transitivity, where edges are forced to share many triangles. The class a network belongs to changes its percolation properties. For networks with weak clustering, we find analytically the critical point for the onset of the giant component and its size. By means of numerical simulations, we also prove that, when comparing with the unclustered counterpart, weak clustering hinders the onset of the giant connected component whereas it is favored by strong clustering. This is a direct consequence of the differences in the k-core structure for the two types of networks. In the particular case of scale-free networks, and although clustering can strongly affect the size and the resilience of the giant connected component, neither weak nor strong transitivity can restore a finite percolation threshold which, in turn, implies the absence of an epidemic threshold. [Preview Abstract] |
Wednesday, March 7, 2007 4:18PM - 4:54PM |
S7.00004: Critical 2-D Percolation: Crossing Probabilities, Modular Forms and Factorization Invited Speaker: We first consider crossing probabilities in critical 2-D percolation in rectangular geometries, derived via conformal field theory. These quantities are shown to exhibit interesting modular behavior [1], although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension $1/3$), follows from a simple modular argument. \\ We next consider the probability of crossing between various points for percolation in the upper half-plane. For two points, with the point $x$ an edge of the system, the probability is \begin{equation} \nonumber {\cal P}(x,z)= k \frac1{y^{5/48}} \Phi(x,z)^{1/3} \end{equation} where $\Phi$ is the potential at $z$ of a 2-D dipole located at $x$, and $k$ is a non-universal constant. For three points, one finds the exact and universal factorization [2,3] \begin{equation} \nonumber {\cal P}(x_1,x_2,z)= C \; \sqrt{{\cal P}(x_1,z){\cal P}(x_2,z){\cal P}(x_1,x_2)} \end{equation} with \begin{equation}\nonumber C= \frac{8 \sqrt{2}\; \pi^{5/2}}{3^{3/4} \; \Gamma(1/3)^{9/2}}. \end{equation} These results are calculated by use of conformal field theory. Computer simulations verify them very precisely. Furthermore, simulations show that the same factorization holds asymptotically, with the same value of $C$, when one or both of the points $x_i$ are moved from the edge into the bulk.\\ 1.) Peter Kleban and Don Zagier, Crossing probabilities and modular forms, J. Stat. Phys. 113, 431-454 (2003) [arXiv: math-ph/0209023].\\ 2.) Peter Kleban, Jacob J. H. Simmons, and Robert M. Ziff, Anchored critical percolation clusters and 2-d electrostatics, Phys. Rev. Letters 97,115702 (2006) [arXiv: cond-mat/0605120].\\ 3.) Jacob J. H. Simmons and Peter Kleban, in preparation. [Preview Abstract] |
Wednesday, March 7, 2007 4:54PM - 5:30PM |
S7.00005: Quantum phase transitions on percolating lattices Invited Speaker: When a quantum many-particle system exists on a randomly diluted lattice, its intrinsic thermal and quantum fluctuations coexist with geometric fluctuations due to percolation. In this talk, we explore how the interplay of these fluctuations influences the phase transition at the percolation threshold. While it is well known that thermal fluctuations generically destroy long-range order on the critical percolation cluster, the effects of quantum fluctuations are more subtle. In diluted quantum magnets with and without dissipation, this leads to novel universality classes for the zero-temperature percolation quantum phase transition. Observables involving dynamical correlations display nonclassical scaling behavior that can nonetheless be determined exactly in two dimensions. Moreover, by exploring a relation between quantum Hamiltonians and classical nonequilibrium processes, we demonstrate that exotic percolation transitions can also occur in epidemic spreading and diffusion-limited chemical reactions. [Preview Abstract] |
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