Z-Vortex Percolation in the Electroweak Crossover Region

We study the statistical properties of Z-vortices and Nambu monopoles in the 3D SU(2) Higgs model for a Higgs mass M_H \approx 100 GeV near and above the crossover temperature, where these defects are thermally excited. Although there is no phase transition at that strong selfcoupling, we observe that the Z-vortices exhibit the percolation transition that has been found recently to accompany the first order thermal transition that exists at smaller Higgs mass. Above the crossover temperature percolating networks of Z-vortex lines are ubiquitous, whereas vortices form a dilute gas of closed vortex loops and (Nambu) monopolium states on the low-temperature side of the crossover. The percolation temperature turns out to be roughly independent of the lattice spacing. We find that the Higgs modulus is smaller (the gauge action is larger) inside the vortices, compared to the bulk average. This correlation becomes very strong on the low-temperature side. The percolation transition is a prerequisite of some string mediated baryon number generation scenarios.Comment: 16 pages, LaTeX, 12 figures, epsf.sty needed; final version to appear in Phys. Lett.

Concurrent enhancement of percolation and synchronization in adaptive networks

Co-evolutionary adaptive mechanisms are not only ubiquitous in nature, but also beneficial for the functioning of a variety of systems. We here consider an adaptive network of oscillators with a stochastic, fitness-based, rule of connectivity, and show that it self-organizes from fragmented and incoherent states to connected and synchronized ones. The synchronization and percolation are associated to abrupt transitions, and they are concurrently (and significantly) enhanced as compared to the non-adaptive case. Finally we provide evidence that only partial adaptation is sufficient to determine these enhancements. Our study, therefore, indicates that inclusion of simple adaptive mechanisms can efficiently describe some emergent features of networked systems' collective behaviors, and suggests also self-organized ways to control synchronization and percolation in natural and social systems.Comment: Published in Scientific Report

A lower bound on the critical parameter of interlacement percolation in high dimension

We investigate the percolative properties of the vacant set left by random interlacements on Z^d, when d is large. A non-negative parameter u controls the density of random interlacements on Z^d. It is known from arXiv:0704.2560, and arXiv:0808.3344, that there is a non-degenerate critical value u_*, such that the vacant set at level u percolates when u < u_*, and does not percolate when u > u_*. Little is known about u_*, however for large d, random interlacements on Z^d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, for which the corresponding critical parameter can be explicitly computed, see arXiv:0907.0316. We prove in this article a lower bound on u_*, which is equivalent to log(d) as d goes to infinity. This lower bound is in agreement with the above mentioned heuristics.Comment: 31 pages, 1 figure, accepted for publication in Probability Theory and Related Field

Statistics of correlated percolation in a bacterial community

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.P50 GM085764 - NIGMS NIH HHS; Howard Hughes Medical Institute; R01 GM121888 - NIGMS NIH HHSPublished versio

Applications of percolation theory to fungal spread with synergy

There is increasing interest in the use of the percolation paradigm to analyze and predict the progress of disease spreading in spatially-structured populations of animals and plants. The wider utility of the approach has been limited, however, by several restrictive assumptions, foremost of which is a strict requirement for simple nearest-neighbour transmission, in which the disease history of an individual is in uenced only by that of its neighbours. In a recent paper the percolation paradigm has been generalised to incorporate synergistic interactions in host infectivity and susceptibility and the impact of these interactions on the invasive dynamics of an epidemic has been demonstrated. In the current paper we elicit evidence that such synergistic interactions may underlie transmission dynamics in real-world systems by rst formulating a model for the spread of a ubiquitous parasitic and saprotrophic fungus through replicated populations of nutrient sites and subsequently tting and testing the model using data from experimental microcosms. Using Bayesian computational methods for model tting, we demonstrate that synergistic interactions are necessary to explain the dynamics observed in the replicate experiments. The broader implications of this work in identifying disease control strategies that de ect epidemics from invasive to non-invasive regimes are discussed