  The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.00169,
2018), is the space of graph parameters that are additive under disjoint union,
multiplicative under the strong product, normalized and monotone under
homomorphisms between the complements. He used it to obtain a dual
characterization of the Shannon capacity of graphs as the minimum of the
evaluation function over the asymptotic spectrum and noted that several known
upper bounds, including the Lov\'asz number and the fractional Haemers bounds
are in fact elements of the asymptotic spectrum (spectral points).
  We show that every spectral point admits a probabilistic refinement and
characterize the functions arising in this way. This reveals that the
asymptotic spectrum can be parameterized with a convex set and the evaluation
function at every graph is logarithmically convex. One consequence is that for
any incomparable pair of spectral points $f$ and $g$ there exists a third one
$h$ and a graph $G$ such that $h(G)<\min\{f(G),g(G)\}$, thus $h$ gives a better
upper bound on the Shannon capacity of $G$. In addition, we show that the
(logarithmic) probabilistic refinement of a spectral point on a fixed graph is
the entropy function associated with a convex corner.
