Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps

We study the positive recurrence of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. We first provide a generic method to construct a Lyapunov function when the drift can be extended to a smooth function on $\mathbb R^N$, using an associated deterministic dynamical system. This approach gives an elementary proof of ergodicity without needing to establish the convergence of the scaled version of the process towards a fluid limit and then proving that the stability of the fluid limit implies the stability of the process. We also provide a counterpart result proving instability conditions. We then show how discontinuous drifts change the nature of the stability conditions and we provide generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piece-wise constant drifts in dimension 2.Comment: 18 pages, 4 figure

An electronic Mach-Zehnder interferometer in the Fractional Quantum Hall effect

We compute the interference pattern of a Mach-Zehnder interferometer operating in the fractional quantum Hall effect. Our theoretical proposal is inspired by a remarkable experiment on edge states in the Integer Quantum Hall effect (IQHE). The Luttinger liquid model is solved via two independent methods: refermionization at nu=1/2 and the Bethe Ansatz solution available for Laughlin fractions. The current differs strongly from that of single electrons in the strong backscattering regime. The Fano factor is periodic in the flux, and it exhibits a sharp transition from sub-Poissonian (charge e/2) to Poissonian (charge e) in the neighborhood of destructive interferences

Polarized heat current generated by quantum pumping in two-dimensional topological insulators

We consider transport properties of a two dimensional topological insulator in a double quantum point contact geometry in presence of a time-dependent external field. In the proposed setup an external gate is placed above a single constriction and it couples only with electrons belonging to the top edge. This asymmetric configuration and the presence of an ac signal allow for a quantum pumping mechanism, which, in turn, can generate finite heat and charge currents in an unbiased device configuration. A microscopic model for the coupling with the external time-dependent gate potential is developed and the induced finite heat and charge currents are investigated. We demonstrate that in the non-interacting case, heat flow is associated with a single spin component, due to the helical nature of the edge states, and therefore a finite and polarized heat current is obtained in this configuration. The presence of e-e interchannel interactions strongly affects the current signal, lowering the degree of polarization of the system. Finally, we also show that separate heat and charge flows can be achieved, varying the amplitude of the external gate.Comment: 13 pages, 5 figure

Weak-field Hall effect and static polarizability of Bloch electrons

A theory of the weak field Hall effect of Bloch electrons based on the analysis of the forces acting on electrons is presented. It is argued that the electric current is composed of two contributions, that driven by the electric field along current flow and the non-dissipative contribution originated in demagnetization currents. The Hall resistance as a function of the electron concentration for the tight-binding model of a crystal with square lattice and body-centered cubic lattice is described in detail. For comparison the effect of strong magnetic fields is also discussed

Stochastic bounds for two-layer loss systems

This paper studies multiclass loss systems with two layers of servers, where each server at the first layer is dedicated to a certain customer class, while the servers at the second layer can handle all customer classes. The routing of customers follows an overflow scheme, where arriving customers are preferentially directed to the first layer. Stochastic comparison and coupling techniques are developed for studying how the system is affected by packing of customers, altered service rates, and altered server configurations. This analysis leads to easily computable upper and lower bounds for the performance of the system

Euclidean versus hyperbolic congestion in idealized versus experimental networks

This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum length routing. More specifically, it is shown that, in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is independent of the radius R whereas, in a Euclidean ball, the same proportion scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as the square of the volume, whereas the same traffic load scales as the volume to the power (n+1)/n in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel data base and synthetic Euclidean lattice networks. It is further conjectured that for networks that do not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of maximum traffic is near the center of mass of the network.Comment: 23 pages, 4 figure

The Large Scale Curvature of Networks

Understanding key structural properties of large scale networks are crucial for analyzing and optimizing their performance, and improving their reliability and security. Here we show that these networks possess a previously unnoticed feature, global curvature, which we argue has a major impact on core congestion: the load at the core of a network with N nodes scales as N^2 as compared to N^1.5 for a flat network. We substantiate this claim through analysis of a collection of real data networks across the globe as measured and documented by previous researchers.Comment: 4 pages, 5 figure