We put forward a general field theory for membranes with embedded activators and analyse their critical properties using renormalization group techniques. Depending on the membrane-activator coupling, we find a crossover between acoustic and diffusive scaling regimes, with mean-field dynamical critical exponents z = 1 and 2 respectively. We argue that the acoustic scaling, which is exact in all spatial dimensions, is a suitable candidate for the universal description of the spatiotemporal patterns observed at the leading edge of motile cells. Furthermore, one-loop corrections to the diffusive mean-field exponents reveal universal behaviour distinct from the Kardar-Parisi-Zhang scaling of passive interfaces and signs of strong-coupling behaviour.
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