Quenched Decay of Correlations for One-dimensional Random Lorenz Maps

We study rates of mixing for small random perturbations of one-dimensional Lorenz maps. Using a random tower construction, we prove that, for Hölder observables, the random system admits exponential rates of quenched correlation decay.

Competitive Equilibrium with Indivisible Goods and Generic Budgets

Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods (Foley 1967, Varian 1974). Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible even in the simple two-agent, single-item market. Yet it is easy to see that, once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper, we aim to extend this approach beyond the single-item case and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that, for agents with equal budgets, making the budgets generic—by adding vanishingly small random perturbations—ensures the existence of equilibrium. We further consider agents with arbitrary nonequal budgets, representing nonequal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among nonequal agents and that it exists in cases of interest (such as when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of nonequals.

On classes of infinite loaded graphs with randomly deleted edges

For a collection of infinite loaded graphs, random perturbations of special type are considered. It is shown that some known classes of these graphs are stable with respect to small random perturbations of this type, while the rest are not.

Young bilateral supernova remnants evolving into a turbulent interstellar magnetic field

ABSTRACT We employ 3D magnetohydrodynamic simulations to study the morphology and synchrotron emission of young supernova remnants evolving in a turbulent interstellar magnetic field, seeking to shed new light on to the polarization structure of the emission and on the debate concerning the quasi-parallel and quasi-perpendicular acceleration mechanisms. In the simulations, we consider a non-homogeneous interstellar medium magnetic field by introducing small random perturbations in the direction and intensity of the field. In order to analyse the dependence of the radio morphology on the degree of magnetic field perturbation and the observer’s point of view, we compute synthetic maps of the polarized intensity, position-angle, polarization fraction, and the polar-reference angle. By comparing the distribution of this angle to the polarization intensity, we show that it is possible to identify what type of acceleration mechanism is taking place at the main shock front.

Random Dynamical Systems Generated by Two Allee Maps

In this paper, we study random dynamical systems generated by two Allee maps. Two models are considered — with and without small random perturbations. It is shown that the behavior of the systems is very similar to the behavior of the deterministic system if we use strictly increasing Allee maps. However, in the case of unimodal Allee maps, the behavior can dramatically change irrespective of the initial conditions.