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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 underhomomorphisms between the complements. He used it to obtain a dualcharacterization of the Shannon capacity of graphs as the minimum of theevaluation function over the asymptotic spectrum and noted that several knownupper bounds, including the Lov\'asz number and the fractional Haemers boundsare in fact elements of the asymptotic spectrum (spectral points).  We show that every spectral point admits a probabilistic refinement andcharacterize the functions arising in this way. This reveals that theasymptotic spectrum can be parameterized with a convex set and the evaluationfunction at every graph is logarithmically convex. One consequence is that forany 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 betterupper bound on the Shannon capacity of $G$. In addition, we show that the(logarithmic) probabilistic refinement of a spectral point on a fixed graph isthe entropy function associated with a convex corner.

Probabilistic refinement of the asymptotic spectrum of graphs