Dichromatic polynomials and Potts models summed over rooted maps

Details Dichromatic polynomials and Potts models summed over rooted maps Dichromatic polynomials and Potts models summed over rooted maps

2000-11-23

arxiv.org/abs/cond-mat/0011400v2

Ann.Comb. 5 (2001) 17-36

Physics

Condensed Matter

Statistical Mechanics

29 pages, three figures changes in App D, introduction and acknowledgements

Scientific paper

10.1016/S0030-4018(00)01091-9

We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the $q$-state Potts model randomized over such maps. Like the regular ferromagnetic lattice models, it has a first-order transition when $q$ is greater than a critical value $q_c$, but $q_c$ is much larger - about 72 instead of 4.

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