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The Q-state Potts model can be extended to noninteger and even complex Q inthe FK representation. In the FK representation the partition function,Z(Q,a),is a polynomial in Q and v=a-1(a=e^-T) and the coefficients of thispolynomial,Phi(b,c), are the number of graphs on the lattice consisting of bbonds and c connected clusters. We introduce the random-cluster transfer matrixto compute Phi exactly on finite square lattices. Given the FK representationof the partition function we begin by studying the critical Potts modelZ_{CP}=Z(Q,a_c), where a_c=1+sqrt{Q}. We find a set of zeros in the complexw=sqrt{Q} plane that map to the Beraha numbers for real positive Q. We alsoidentify tilde{Q}_c(L), the value of Q for a lattice of width L above which thelocus of zeros in the complex p=v/sqrt{Q} plane lies on the unit circle. Wefind that 1/tilde{Q}_c->0 as 1/L->0. We then study zeros of the AF Potts modelin the complex Q plane and determine Q_c(a), the largest value of Q for a fixedvalue of a below which there is AF order. We find excellent agreement withQ_c=(1-a)(a+3). We also investigate the locus of zeros of the FM Potts model inthe complex Q plane and confirm that Q_c=(a-1)^2. We show that the edgesingularity in the complex Q plane approaches Q_c as Q_c(L)~Q_c+AL^-y_q, anddetermine the scaling exponent y_q. Finally, by finite size scaling of theFisher zeros near the AF critical point we determine the thermal exponent y_tas a function of Q in the range 2<Q<3. We find that y_t is a smooth function ofQ and is well fit by y_t=(1+Au+Bu^2)/(C+Du) where u=u(Q). For Q=3 we findy_t~0.6; however if we include lattices up to L=12 we find y_t~0.50.Comment: to appear in Physical Review

Density of states, Potts zeros, and Fisher zeros of theQ-state Potts model for continuousQ