Classical particle in a box with random potential: Exploiting rotational symmetry of replicated Hamiltonian

We provide a detailed discussion of the replica approach to thermodynamics of a single classical particle placed in a random Gaussian N(≫1)N(≫1)-dimensional potential inside a spherical box of a finite radius L=RN. Earlier solutions of R=∞R=∞ version of this model were based on applying the Gaussian Variational Ansatz (GVA) to the replicated partition function, and revealed a possibility of glassy phases at low temperatures. For a general R  , we show how to utilize instead the underlying rotational symmetry and to arrive to a compact expression for the free energy in the limit N→∞N→∞ directly, without any need for intermediate variational approximations. This method reveals a striking similarity with the much-studied spherical model of spin glasses. Depending on the competition between the radius R   and the curvature of the parabolic confining potential μ⩾0μ⩾0, as well as on the three types of disorder—short-ranged, long-ranged, and logarithmic—the phase diagram of the system in the (μ,T)(μ,T) plane undergoes considerable modifications. In the limit of infinite confinement radius our analysis confirms all previous results obtained by GVA. The paper has also a considerable pedagogical component by providing an extended presentation of technical details which are not always easy to find in the existing literature.