■ - ita’s diary

Exercise 1: Estimate the critical exponent $\alpha$ of percolation in three dimensions using numerical simulations.

Solution: Let us denote the number of states which has b connected bonds and c clusters by N(b,c). Through the Fortuin-Kasteleyn transformation, the partition function of the Q-state Potts model can be witten down as $Z(Q)=\sum_{b,c} p^b(1-p)^{V-b} Q^c N(b,c) \sim \sum_{b,c} N(b,c) exp ( b \epsilon + c log Q)$ , where $\epsilon = log p/(1-p)$ .
The specific heat is then proportional to $(\partial / \partial \epsilon)^2 log Z(Q, \epsilon)$ . When Q=1, $Z(Q,\epsilon )$ shows no singular behaviour. But this should be interpreted that the coeficient of the singular part becomes zero when Q=1 and the exponent $\alpha$ should be calculated in the $Q\rightarrow 1$ limit. Thus the quantity to observe is $(\partial / \partial q) (\partial / \partial \epsilon)^2 \log Z(1+q, \epsilon)|_{q=0}=\langle (b-\langle b \rangle)^2 (c - \langle c \rangle)\rangle$ . The value of this quantity at Pc behaves as const $+ a L^{\alpha/\nu}$ .