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Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

We consider the random-bond Potts model in the large-$Q$ limit and calculate the excess entropy, $S_Γ$, of a contour, $Γ$, which is given by the mean number of Fortuin-Kasteleyn clusters which are crossed by $Γ$. In two dimensions $S_Γ$ is proportional to the length of $Γ$, to which - at the critical point - there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: $\lim_{Q \to \infty}c(Q)/\ln Q =0.74(2)$, close to previous estimates calculated at finite values of $Q$.