  We perform Monte Carlo simulations of a four-dimensional gauge invariant spin
system which describes random surfaces with gonihedric action. We develop the
analogy between the flat-crumpled phase transition of the lattice surface model
and the liquid-gas phase transition of non-ideal gases, and identify the
self-intersection coupling constant $k$ of the surface model with the pressure
$P$. As $k$ increases the system moves to a critical point in complete analogy
with the situation for non-ideal gases, where the liquid and the gas phases
approach each other with increasing $P$. We measure vacuum expectation values
of various operators and the corresponding critical indices.
