Renyi entropy of a line in two-dimensional Ising models

We consider the two-dimensional Ising model on an infinitely long cylinder and study the probabilities pi to observe a given spin configuration i along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave functions. We analyze the subleading constant to the Renyi entropy R-n=1/(1-n)ln (Sigma(i)p(i)(n)) and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a steplike fashion as a function of n with a discontinuity at the Shannon point n=1. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the Renyi parameter are of special interest: n=1/2 and n=infinity are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions, respectively.