  In this work, we study the probability distribution for the force and
potential energy of a test particle interacting with $N$ point random sources
in the limit $N\rightarrow\infty$. The interaction is given by a central
potential $V(R)=k/R^{\delta-1}$ in a $ d$-dimensional euclidean space, where
$R$ is the random relative distance between the source and the test particle,
$\delta$ is the force exponent, and $k$ is the coupling parameter. In order to
assure a well-defined limit for the probability distribution of the force and
potential energy, we { must} renormalize the coupling parameter and/or the
system size as a function of the number $N$ of sources.
  We show the existence of three non-singular limits, depending on the exponent
$\delta$ and the spatial dimension $d$. (i) For $\delta<d$ the force and
potential energy { converge} to their respective mean values. This limit is
called Mean Field Limit. (ii) For $\delta>d+1$ the potential energy converges
to a random variable and the force to a random vector. This limit is called
Thermodynamic Limit. (iii) For $d<\delta<d+1$ the potential energy converges to
its mean and the force to a random vector. This limit is called Mixed Limit
  Also, we show the existence of two singular limits: (iv) for $\delta=d$ the
potential energy converges to its mean and the force to zero, and (v) for
$\delta=d+1$ the energy converges to a finite value and the force to a random
vector.
