The dynamics of critical fluctuations in asymmetric Curie–Weiss models

Delay-Induced Periodic Behaviour in Competitive Populations

“…Then the conclusion follows as in Proposition 2.1 taking G d the generator of the semigroup associated to the deterministic evolution (8).…”

“…Also, we are interested in how perturbations of different types may change the large scale picture of the system as the number of agents grows to infinity, giving rise to periodic self-organized behaviours of the agents. This type of phenomenon has already been described in many models of spin systems with mean-field interaction where some kind of frustration is present in the agents' attitude; for example, in [3,8,9], a dissipation term (that, in absence of interaction, leads the system to a neutral condition where no action is preferred over the other) is added in the evolution of Curie-Weiss-like models.…”

“…where v denotes the norm of the vector v. Then, as N → +∞, the sequence of Markov processes (m N , γ N , η N ); N ≥ 1 converges in distribution, with respect to the Skorohod topology, to the unique solution of the system of equations (8) with m(0), γ (0), η(0) = ( m, γ , η).…”

Path-space moderate deviations for a class of Curie–Weiss models with dissipation

“…Instead of considering the usual radius and angle variables, we are using the pair radius squared and angle. This is for notational convenience and to be consistent with notation in [14]. Moreover, for any fixed radius, the parametric curve obtained in R 2 is traversed clockwise, rather than anticlockwise, while increasing the angle in S 1 .…”

“…it shows a time periodic behaviour despite of the fact that no periodic force is applied. More recently, fluctuations on the level of a path-space (standard and non-standard) central limit theorem for the noiseless version of the same model were studied in [14]. Our aim is to continue the analysis of fluctuations by characterizing their dynamical features whenever looking for moderate size deviations from the average value.…”