Varied phenomenology of models displaying dynamical large-deviation singularities

Abstract

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

Delocalization-localization dynamical phase transition of random walks on graphs

“…We make use of a theoretical framework for the calculation of large deviations that we developed in [18] and that allows us to: (i) consider general time-additive observables, (ii) analytically characterize the behavior of random walks on finite-size graphs, and (iii) rigorously study the scaling (with respect to the size of the graph) of fluctuations around the critical value of the DPT. Remarkably, in agreement with [22,23] we notice that an important ingredient for the appearance of a first order DPT in both models is the presence of absorbing dynamics, generated by different scalings of the hopping probabilities in the graph. Furthermore, we notice that although the first order DPT appears in both the models we investigated, the scaling of the fluctuations around the transition is different and we argue that it is both function of the dynamical process and of the inherent topology of the network.…”

“…There are good grounds to consider it as a first-order DPT where we observe the coexistence of two 'phases' characterized by random walk paths that visit the whole graph, and paths localized in dangling chains, i.e., lowly connected structures of the graph. However, a rigorous proof for ensembles of random graphs is still lacking and, in fact, the community still debates on the real nature and interpretation of DPTs [22,23].…”

Dynamical Large Deviations of Two-Dimensional Kinetically Constrained Models Using a Neural-Network State Ansatz

“…The large-deviation properties of the one-dimensional FA model are well studied. In the limit of large system size there exists a singularity in the SCGF of the activity at a size-dependent value of s. Singularities in the SCGF are often associated with phase transitions-in this case a dynamical phase transition between an active and an inactive phase [2,30,[40][41][42]though this is not always the case [43]. In what follows, we show that a neural-network state ansatz can determine the scaling behavior of similar large-deviation singularities in a two-dimensional kinetically constrained model, and can describe the spatial correlations of trajectories displaying atypically large activity.…”

Cellular automata can classify data by inducing trajectory phase coexistence

“…The bimodal dynamical behavior shown in Fig. 2(a) is similar to that seen in systems with explicit feedback or memory [29,37], and resembles an emergent version of a spiking neuron [20,21]. Spiking neurons are timedependent models of biological neurons, often modeled by differential equations, whose outputs can vary sharply with time in response to a stimulus.…”