  We study the Shannon entropy of the probability distribution resulting from
the ground-state wave function of a one-dimensional quantum model. This entropy
is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function
built from the corresponding two-dimensional classical model. In both critical
and massive cases, we observe that it is composed of an extensive part
proportional to the length of the system and a subleading universal constant
S_0. In c=1 critical systems (Tomonaga-Luttinger liquids), we find that S_0 is
a simple function of the boson compactification radius. This finding is based
on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and
Calogero-Sutherland models. We also performed numerical demonstrations in the
dimer models and the spin-1/2 XXZ chain. In a massive (crystal) phase, S_0 is
related to the ground-state degeneracy. We also examine this entropy in the
Ising chain in a transverse field as an example showing a c=1/2 critical point.
