  An $\al$-permanental process $\{X_{ t},t\in T \}$ is a stochastic process
determined by a kernel $K=\{K(s,t),s,t\in T \}$, with the property that for all
$t_{1},\ldots,t_{n}\in T $, $ |I+K( t_{1},\ldots,t_{n}) S|^{- \al} $ is the
Laplace transform of $(X_{t_{1}},\ldots,X_{t_{n}})$, where $ K(
t_{1},\ldots,t_{n})$ denotes the matrix $\{K(t_{i}, t_{j})\}_{i,j=1}^{n}$ and
$S$ is the diagonal matrix with entries $s_{1},\ldots,s_{n} $. $
(X_{t_{1}},\ldots,X_{t_{n}})$ is called a permanental vector.
  Under the condition that $K$ is the potential density of a transient Markov
process,
  $(X_{t_{1}},\ldots,X_{t_{n}})$ is represented as a random mixture of
  $n$-dimensional random variables with components that are independent gamma
random variables. This representation leads to a Sudakov type inequality for
the sup-norm of $(X_{t_{1}},\ldots,X_{t_{n}})$ that is used to obtain
sufficient conditions for a large class of permanental processes to be
unbounded almost surely. These results are used to obtain conditions for
permanental processes associated with certain L\'evy processes to be unbounded.
  Because $K$ is the potential density of a transient Markov process, for all
$t_{1},\ldots,t_{n}\in T $, $A( t_{1},\ldots,t_{n}):= (K(
t_{1},\ldots,t_{n}))^{-1}$ are $M$-matrices. The results in this paper are
obtained by working with these $M$-matrices.
