Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian

We investigate thermodynamics of a single classical particle placed in a spherical box of a finite radius $R$ and subject to a superposition of a $N-$dimensional Gaussian random potential and the parabolic potential with the curvature $\mu>0$. Earlier solutions of $R\to \infty$ version of this model were based on combining the replica trick with the Gaussian Variational Ansatz (GVA) for free energy, and revealed a possibility of a glassy phase at low temperatures. For a general $R$, we show how to utilize instead the underlying rotational symmetry of the replicated partition function and to arrive to a compact expression for the free energy in the limit $N\to \infty$ directly, without any need for intermediate variational approximations. This method reveals striking similarity with the much-studied spherical model of spin glasses. Depending on the value of $R$ and the three types of disorder - short-ranged, long-ranged, and logarithmic - the phase diagram of the system in the $(\mu,T)$ plane undergoes considerable modifications. In the limit of infinite confinement radius our analysis confirms all previous results obtained by GVA.