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We study the q-state Potts model with nearest-neighbor coupling v=e^{\betaJ}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially,this limit gives rise to the generating polynomial of spanning forests;physically, it provides information about the Potts-model phase diagram in theneighborhood of (q,v) = (0,0). We have studied this model on the square andtriangular lattices, using a transfer-matrix approach at both real and complexvalues of w. For both lattices, we have computed the symbolic transfer matricesfor cylindrical strips of widths 2 \le L \le 10, as well as the limiting curvesof partition-function zeros in the complex w-plane. For real w, we find twodistinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp.w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w >w_0 we find a non-critical disordered phase, while for w < w_0 our results arecompatible with a massless Berker-Kadanoff phase with conformal charge c = -2and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w =w_0 we find a "first-order critical point": the first derivative of the freeenergy is discontinuous at w_0, while the correlation length diverges as w\downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0seems to be the same for both lattices and it differs from that of theBerker-Kadanoff phase: our results suggest that the conformal charge is c = -1,the leading thermal scaling dimension is x_{T,1} = 0, and the criticalexponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65  Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and  forests_tri_2-9P.m. Final journal versio

Spanning Forests and the q-State Potts Model in the Limit q →0