  We study the statistical mechanics of a single active slider on a fluctuating
interface, by means of numerical simulations and theoretical arguments. The
slider, which moves by definition towards the interface minima, is active as it
also stimulates growth of the interface. Even though such a particle has no
counterpart in thermodynamic systems, active sliders may provide a simple model
for ATP-dependent membrane proteins that activate cytoskeletal growth. We find
a wide range of dynamical regimes according to the ratio between the timescales
associated with the slider motion and the interface relaxation. If the
interface dynamics is slow, the slider behaves like a random walker in a random
envinronment which, furthermore, is able to escape environmental troughs by
making them grow. This results in different dynamic exponens to the interface
and the particle: the former behaves as an Edward-Wilkinson surface with
dynamic exponent 2 whereas the latter has dynamic exponent 3/2. When the
interface is fast, we get sustained ballistic motion with the particle surfing
a membrane wave created by itself. However, if the interface relaxes
immediately (i.e., it is infinitely fast), particle motion becomes symmetric
and goes back to diffusive. Due to such a rich phenomenology, we propose the
active slider as a toy model of fundamental interest in the field of active
membranes and, generally, whenever the system constituent can alter the
environment by spending energy.
