We consider a directed percolation process at its critical point. The probability that the deviation of the global order parameter with respect to its average has not changed its sign between 0 and t decays with t as a power law. In space dimensions d greater than or equal to 4 the global persistence exponent theta(p) that characterizes this decay is theta(p) = 2 while for d < 4 its value is increased to first order in epsilon = 4-d. Combining a method developed by Majumdar and Sire with renormalization group techniques we compute the correction to theta(p) to first order in epsilon. The global persistence exponent is found to be a new and independent exponent. Finally we compare our results with existing simulations. [References: 13]
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