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The order of a phase transition is usually determined by the nature of thesymmetry breaking at the phase transition point and the dimension of the modelunder consideration. For instance, q-state Potts models in two dimensionsdisplay a second order, continuous transition for q = 2,3,4 and first order forhigher q.  Tamura et al recently introduced Potts models with "invisible" states whichcontribute to the entropy but not the internal energy and noted that addingsuch invisible states could transmute continuous transitions into first ordertransitions. This was observed both in a Bragg-Williams type mean-fieldcalculation and 2D Monte-Carlo simulations. It was suggested that the invisiblestate mechanism for transmuting the order of a transition might play a rolewhere transition orders inconsistent with the usual scheme had been observed.  In this paper we note that an alternative mean-field approach employing3-regular random ("thin") graphs also displays this change in the order of thetransition as the number of invisible states is varied, although the number ofstates required to effect the transmutation, 17 invisible states when there are2 visible states, is much higher than in the Bragg-Williams case. Thecalculation proceeds by using the equivalence of the Potts model with 2 visibleand r invisible states to the Blume-Emery-Griffiths (BEG) model, so aby-product is the solution of the BEG model on thin random graphs.

Potts Models with (17) Invisible States on Thin Graphs