Impact of string and monopole-type junctions on domain wall dynamics: implications for dark energy

We investigate the potential role of string and monopole-type junctions in the frustration of domain wall networks using a velocity-dependent one-scale model for the characteristic velocity, $v$, and the characteristic length, $L$, of the network. We show that, except for very special network configurations, v^2 \lsim (HL)^2 \lsim (\rho_\sigma + \rho_\mu)/\rho_m where $H$ is the Hubble parameter and $\rho_\sigma$, $\rho_\mu$ and $\rho_m$ are the average density of domain walls, strings and monopole-type junctions. We further show that if domain walls are to provide a significant contribution to the dark energy without generating exceedingly large CMB temperature fluctuations then, at the present time, the network must have a characteristic length L_0 \lsim 10 \Omega_{\sigma 0}^{-2/3} {\rm kpc} and a characteristic velocity v_0 \lsim 10^{-5} \Omega_{\sigma 0}^{-2/3} where $\Omega_{\sigma 0}=\rho_{\sigma 0}/\rho_{c 0}$ and $\rho_c$ is the critical density. In order to satisfy these constraints with $\Omega_{\sigma 0} \sim 1$, $\rho_{m 0}$ would have to be at least 10 orders of magnitude larger than $\rho_{\sigma 0}$, which would be in complete disagreement with observations. This result provides very strong additional support for the conjecture that no natural frustration mechanism, which could lead to a significant contribution of domain walls to the dark energy budget, exists.Comment: 4 pages, 1 figur

The Immunity of Polymer-Microemulsion Networks

The concept of network immunity, i.e., the robustness of the network connectivity after a random deletion of edges or vertices, has been investigated in biological or communication networks. We apply this concept to a self-assembling, physical network of microemulsion droplets connected by telechelic polymers, where more than one polymer can connect a pair of droplets. The gel phase of this system has higher immunity if it is more likely to survive (i.e., maintain a macroscopic, connected component) when some of the polymers are randomly degraded. We consider the distribution $p(\sigma)$ of the number of polymers between a pair of droplets, and show that gel immunity decreases as the variance of $p(\sigma)$ increases. Repulsive interactions between the polymers decrease the variance, while attractive interactions increase the variance, and may result in a bimodal $p(\sigma)$.Comment: Corrected typo

Storing cycles in Hopfield-type networks with pseudoinverse learning rule: admissibility and network topology

Cyclic patterns of neuronal activity are ubiquitous in animal nervous systems, and partially responsible for generating and controlling rhythmic movements such as locomotion, respiration, swallowing and so on. Clarifying the role of the network connectivities for generating cyclic patterns is fundamental for understanding the generation of rhythmic movements. In this paper, the storage of binary cycles in neural networks is investigated. We call a cycle $\Sigma$ admissible if a connectivity matrix satisfying the cycle's transition conditions exists, and construct it using the pseudoinverse learning rule. Our main focus is on the structural features of admissible cycles and corresponding network topology. We show that $\Sigma$ is admissible if and only if its discrete Fourier transform contains exactly $r={rank}(\Sigma)$ nonzero columns. Based on the decomposition of the rows of $\Sigma$ into loops, where a loop is the set of all cyclic permutations of a row, cycles are classified as simple cycles, separable or inseparable composite cycles. Simple cycles contain rows from one loop only, and the network topology is a feedforward chain with feedback to one neuron if the loop-vectors in $\Sigma$ are cyclic permutations of each other. Composite cycles contain rows from at least two disjoint loops, and the neurons corresponding to the rows in $\Sigma$ from the same loop are identified with a cluster. Networks constructed from separable composite cycles decompose into completely isolated clusters. For inseparable composite cycles at least two clusters are connected, and the cluster-connectivity is related to the intersections of the spaces spanned by the loop-vectors of the clusters. Simulations showing successfully retrieved cycles in continuous-time Hopfield-type networks and in networks of spiking neurons are presented.Comment: 48 pages, 3 figure

Network Models of Quantum Percolation and Their Field-Theory Representations

We obtain the field-theory representations of several network models that are relevant to 2D transport in high magnetic fields. Among them, the simplest one, which is relevant to the plateau transition in the quantum Hall effect, is equivalent to a particular representation of an antiferromagnetic SU(2N) ($N\to 0$) spin chain. Since the later can be mapped onto a $\theta\ne 0$, $U(2N)/U(N)\times U(N)$ sigma model, and since recent numerical analyses of the corresponding network give a delocalization transition with $\nu\approx 2.3$, we conclude that the same exponent is applicable to the sigma model

Gauge Defect Networks in Two-Dimensional CFT

An interpretation of the gauge anomaly of the two-dimensional multi-phase sigma model is presented in terms of an obstruction to the existence of a topological defect network implementing a local trivialisation of the gauged sigma model.Comment: 8 pages; The article is the author's contribution to the Proceedings of the XXIX International Colloquium on Group-Theoretical Methods in Physics (20-26 August 2012, Tianjin, China

First-passage phenomena in hierarchical networks

In this paper we study Markov processes and related first passage problems on a class of weighted, modular graphs which generalize the Dyson hierarchical model. In these networks, the coupling strength between two nodes depends on their distance and is modulated by a parameter $\sigma$. We find that, in the thermodynamic limit, ergodicity is lost and the "distant" nodes can not be reached. Moreover, for finite-sized systems, there exists a threshold value for $\sigma$ such that, when $\sigma$ is relatively large, the inhomogeneity of the coupling pattern prevails and "distant" nodes are hardly reached. The same analysis is carried on also for generic hierarchical graphs, where interactions are meant to involve $p$-plets ($p>2$) of nodes, finding that ergodicity is still broken in the thermodynamic limit, but no threshold value for $\sigma$ is evidenced, ultimately due to a slow growth of the network diameter with the size

Distance Oracles for Time-Dependent Networks

We present the first approximate distance oracle for sparse directed networks with time-dependent arc-travel-times determined by continuous, piecewise linear, positive functions possessing the FIFO property. Our approach precomputes $(1+\epsilon)-$approximate distance summaries from selected landmark vertices to all other vertices in the network. Our oracle uses subquadratic space and time preprocessing, and provides two sublinear-time query algorithms that deliver constant and $(1+\sigma)-$approximate shortest-travel-times, respectively, for arbitrary origin-destination pairs in the network, for any constant $\sigma > \epsilon$. Our oracle is based only on the sparsity of the network, along with two quite natural assumptions about travel-time functions which allow the smooth transition towards asymmetric and time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An extended abstract also appeared in the 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014, track-A

Cosmological Evolution of Global Monopoles

We investigate the cosmological evolution of global monopoles in the radiation dominated (RD) and matter dominated (MD) universes by numerically solving field equations of scalar fields. It is shown that the global monopole network relaxes into the scaling regime, unlike the gauge monopole network. The number density of global monopoles is given by $n(t) \simeq (0.43\pm0.07) / t^{3}$ during the RD era and $n(t) \simeq (0.25\pm0.05) / t^{3}$ during the MD era. Thus, we have confirmed that density fluctuations produced by global monopoles become scale invariant and are given by $\delta \rho \sim 7.2(5.0) \sigma^{2} / t^{2}$ during the RD (MD) era, where $\sigma$ is the breaking scale of the symmetry.Comment: 6 pages, 2 figures, to appear in Phys. Rev. D (R