<p>The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They admit a natural group action, and “random Brody curves” in the title refers to invariant probability measures for it. We study their geometric and dynamical properties. Given an invariant probability measure <i>μ</i> on the space of Brody curves, our first main theorem claims that its rate distortion dimension is bounded by the integral of a “geometric potential” over <i>μ</i>. This result is analogous to the Ruelle inequality of smooth ergodic theory. Our second main theorem claims that there exists a rich variety of invariant probability measures attaining equality in this “Ruelle inequality for Brody curves”. The main tools of the proofs are the deformation theory of Brody curves and the variational principle for mean dimension with potential. This approach is motivated by the theory of thermodynamic formalism for Axiom A diffeomorphisms.</p>