  We consider the geometric random (GR) graph on the $d-$dimensional torus with
the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result
is an exact characterization of the probability that a particular labeled cycle
exists in this random graph. For $\sigma = 2$ and $\sigma = \infty$, we use
this characterization to derive a series which evaluates to the cycle
probability. We thus obtain an exact formula for the expected number of
Hamilton cycles in the random graph (when $\sigma = \infty$ and $\sigma = 2$).
We also consider the adjacency matrix of the random graph and derive a
recurrence relation for the expected values of the elementary symmetric
functions evaluated on the eigenvalues (and thus the determinant) of the
adjacency matrix, and a recurrence relation for the expected value of the
permanent of the adjacency matrix. The cycle probability features prominently
in these recurrence relations. We calculate these quantities for geometric
random graphs (in the $\sigma = 2$ and $\sigma = \infty$ case) with up to $20$
vertices, and compare them with the corresponding quantities for the
Erd\"{o}s-R\'{e}nyi (ER) random graph with the same edge probabilities. The
calculations indicate that the threshold for rapid growth in the number of
Hamilton cycles (as well as that for rapid growth in the permanent of the
adjacency matrix) in the GR graph is lower than in the ER graph. However, as
the number of vertices $n$ increases, the difference between the GR and ER
thresholds reduces, and in both cases, the threshold $\sim \log(n)/n$. Also, we
observe that the expected determinant can take very large values. This throws
some light on the question of the maximal determinant of symmetric $0/1$
matrices.
