Abstract We study the propagation of energy in one-dimensional anharmonic chains subject to a periodic, localized forcing. For the purely harmonic case, forcing frequencies outside the linear spectrum produce exponentially localized responses, preventing equi-distribution of energy per degree of freedom. We extend this result to anharmonic perturbations with bounded second derivatives and boundary dissipation, proving that for small perturbations and non-resonant forcing, the dynamics converges to a periodic stationary state with energy exponentially localized uniformly in the system size. The perturbed periodic state is described by a convergent power type expansion in the strength of the anharmonicity. This excludes chaoticity induced by anharmonicity, independently of the size of the system. Our perturbative scheme can also be applied in higher dimensions.

Abstract We consider the empirical neighborhood distribution of marked sparse Erdős-Rényi random graphs, obtained by decorating edges and vertices of a sparse Erdős-Rényi random graph with i.i.d. random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam (2019) which is inspired on the previous work by Bordenave and Caputo (2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erdős-Rényi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.

Abstract Using molecular dynamics simulations, the ratio of ternary to binary collision densities is investigated for both a hard-sphere gas and a Lennard-Jones gas. The simulations confirm the existing theoretical models for the hard-sphere gas but reveal temperature dependence of collision density ratio in a Lennard-Jones gas. A theoretical model is developed to explain this temperature dependence.

Abstract In this paper, we study the oriented swap process on the positive integers and its asymptotic properties. Our results extend a theorem by Angel, Holroyd, and Romik regarding the trajectories of particles in the finite oriented swap process. Furthermore, we study the evolution of the type of a particle at the leftmost position over time. Our approach relies on a relationship between multi-species particle systems and Hecke algebras, complemented by a detailed asymptotic analysis.

Abstract We provide an analytical framework for analyzing the quality of stochastic Verlet-type integrators for simulating the Langevin equation. Focusing only on basic objective measures, we consider the ability of an integrator to correctly simulate two characteristic configurational quantities of transport, a) diffusion on a flat surface and b) drift on a tilted planar surface, as well as c) statistical sampling of a harmonic potential. For any stochastic Verlet-type integrator expressed in its configurational form, we develop closed form expressions to directly assess these three most basic quantities as a function of the applied time step. The applicability of the analysis is exemplified through twelve representative integrators developed over the past five decades, and algorithm performance is conveniently visualized through the three characteristic measures for each integrator. The GJ set of integrators stands out as the only option for correctly simulating diffusion, drift, and Boltzmann distribution in linear systems, and we therefore suggest that this general method is the one best suited for high quality thermodynamic simulations of nonlinear and complex systems, including for relatively high time steps compared to simulations with other integrators.

Abstract We consider constrained-degree percolation on the hypercubic lattice. This is a continuous-time model defined by a sequence $$(U_e)_{e}$$ ( U e ) e of i.i.d. uniform random variables and a positive integer k , referred to as the constraint. The model evolves as follows: each edge e attempts to open at a random time $$U_e$$ U e , independently of all other edges. It succeeds if, at time $$U_e$$ U e , both of its end-vertices have degrees strictly smaller than k . It is known [21] that this model undergoes a phase transition when $$d\ge 3$$ d ≥ 3 for most nontrivial values of k . In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time $$t\in [0,1)$$ t ∈ [ 0 , 1 ) is almost surely either 0 or 1. We also show that the law of the process is differentiable with respect to time for local events, extending a result of [30]. As a consequence of these two results, we prove that the percolation function is continuous in the supercritical regime $$t\in (t_c,1)$$ t ∈ ( t c , 1 ) , where $$t_c$$ t c denotes the percolation critical threshold. Finally, we show that the two-point connectivity function is bounded away from zero in the supercritical regime.