Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model

Abstract

The East model is the simplest one-dimensional kinetically-constrained model of N spins with a trivial equilibrium that displays anomalously large spatio-temporal fluctuations, with characteristic "space-time bubbles" in trajectory space, and with a discontinuity at the origin for the first derivative of the scaled cumulant generating function of the total activity. These striking dynamical properties are revisited via the large deviations at Level 2.5 for the relevant local empirical densities and flows that only involve two consecutive spins. This framework allows to characterize their anomalous rate functions and to analyze the consequences for all the time-additive observables that can be reconstructed from them, both for the pure and for the random East model. These singularities in dynamical large deviations properties disappear when the hard-constraint of the East model is replaced by the soft constraint.

Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

“…x y =x q(x, y) ln q(x, y) w x,y ρ(y) − q(x, y) + w x,y ρ(y) (47) This rate function characterizes how rare it is for large T to see some empirical density ρ(.) and some empirical flows q(., .)…”

Inhomogeneous asymmetric exclusion processes between two reservoirs: large deviations for the local empirical observables in the mean-field approximation

“…In this section, we follow the approach that has recently been applied to the kineticallyconstrained East model [27] in order to identify the appropriate local empirical observables and to analyze whether it is possible to write closed large deviations properties for them.…”

“…For a given Markov model, the relevant time-empirical observables are defined as the time-empirical observables that determine the trajectories probabilities (see appendix A of [27] for a general discussion). For the present Markov jump process, the probability of equation (A3) for the trajectory C(t) = {S 1 (t), .…”

“…More generally, the local projections of equation (B6) do not give closed constraints for local empirical observables, but produce a whole hierarchy, which is the analog of the hierarchy for the steady state as recalled after equation (11). For many-body dynamics satisfying detailed-balance that converge towards the equilibrium without any steady current, one can use the vanishing of the optimal values of the empirical currents to obtain closed large deviations properties for the local empirical densities and activities, as discussed in detail in [27] with the application to the kinetically-constrained East model. For the present nonequilibrium exclusion models that do not satisfy detailed-balance and that have nonvanishing currents in the steady state, the simplest approximation to close the hierarchy problem is the mean-field approximation for the two-spin empirical density ρ .. ..…”

“…Similarly, for interacting random walkers in the continuous-time discrete-space framework, space-time local Lagrangian have been analyzed in [18,70,71]. In the present paper, we focus on asymmetric exclusion processes with spacedependent rates between two reservoirs and we follow the approach that has recently been applied to the kinetically-constrained East model [27] in order to identify the appropriate local empirical observables and to analyze whether it is possible to write closed large deviations properties for them. Inhomogeneous exclusion processes have been already much studied, either for samples with one or two bottlenecks [72][73][74], or for samples with smoothly-varying hopping rates [75,76], or for disordered samples [77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96].…”