The Cancho i Ferrer - Sol\'e model does not explain Zipf's Law

We examine the cost-minimization problem posed by Ferrer i Cancho and Sol\'e in their information-theory based communication model [1], proposed in efforts to explain Zipf's Law (that is a power-law frequency-rank relation for words in written texts). Using a simple inequality, we obtain the exact minimum-cost solution as a function of the parameter $\lambda$, as obtained previously via other methods [2-4]. ($\lambda$ defines the relative weights of speaker's and listener's costs.) We show that at the phase transition, the minimum-cost solutions do not correspond to a power law except for a vanishingly small subset, even if we impose the additional condition of equal costs to speaker and listener. Finally we consider the model at finite temperature using mean-field theory and entropic Monte Carlo simulation, and find a line of {\it discontinuous} phase transitions in the $\lambda$-$T$ plane. The simulations yield no evidence of a power-law frequency-rank distribution. \noindent 1. R. Ferrer i Cancho and R.~V. Sol\'e, PNAS {\bf 100}, 788 (2003).\\ 2. R. Ferrer i Cancho and A. D\'{\i}az-Guilera, J. Stat. Mech.: Theory Exp. (2007) P06009.\\ 3. A. Trosso, Master's thesis, 2008, University of Turin, Italy.\\ 4. M. Prokopenko et al., J. Stat. Mech. (2010) P11025.