Thermodynamics of histories for the one-dimensional contact process

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The dynamical activity K(t) of a stochastic process is the number of times it changes configuration up to time t. It was recently argued that (spin) glasses are at a first order dynamical transition where histories of low and high activity coexist. We study this transition in the one-dimensional contact process by weighting its histories by exp(sK(t)). We determine the phase diagram and the critical exponents of this model using a recently developed approach to the thermodynamics of histories that is based on the density matrix renormalisation group. We find that for every value of the infection rate, there is a phase transition at a critical value of s. Near the absorbing state phase transition of the contact process, the generating function of the activity shows a scaling behavior similar to that of the free energy in an equilibrium system near criticality.