Delocalization-localization dynamical phase transition of random walks on graphs

Abstract

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a `co-existence' of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

12 pages, 6 figures

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• Statistical physics of social dynamics
• The large deviation approach to statistical mechanics
• Extreme events on complex networks
• Varied phenomenology of models displaying dynamical large-deviation singularities
• Dynamical phase transition to localized states in the two-dimensional random walk conditioned on partial currents
• Random Walks on Complex Networks

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