# The Random-Cluster Model: Corrections

Note: The following corrections have been implemented in the corrected reprint dated 2009 and downloadable via the main page for the book.

The more serious:

• Page 21: Theorem 2.3 is incorrect, with minor consequences for the proofs of Theorems 2.6, 2.19, 3.79: ps pdf
• Page 111: Theorem 5.47 needs a side condition, and Corollary 5.51 has a slip: ps pdf

The less serious:

• Page 28, Example (2.26): Z should be defined as the maximum of X and Y, rather than as their product. That is, Z = X \vee Y.
• Page 53, equation (3.57): The first exponent should be |V| - k(G).
• Page 64, line 2: \Omega_{F,\omega} should be \Omega_{E',\omega}.
• Page 75, Thm 4.19: The superscript b may take the values 0 and 1 throughout.
• Page 95, Thm 4.91 (a) (iv): Replace (4.89) by (4.90).
• Page 157, line -7: The statements about continuity follow as in Theorems 4.63 and 6.17.
• Page 157, line -5: h^0 is automatically left-continuous, and h^1 right-continuous, by Proposition 4.28.
• Page 163: The conjectured equation (6.73) has been verified by Bollobás and Riordan.
• Page 175, equation (7.11): An empty-set character has disappeared from the RHS of the second suffix of the summation.
• Page 176, Figure 7.2: Some of the lines may be incorrectly indicated.
• Page 226, line 22: We can take p_1 \le p_2 and q_1 \ge q_2.
• Page 255, line -4: This statement is of course known for sufficiently large d.
• Page 258, equation (9.3): The first exponent should be |W| - k(H). And in equation (9.5), the second exponent should be |F| - |W| + k(H).
• Page 259: Perhaps E_m and E_P constitutes unfortunate notation.
• Page 260, Thm 9.15: The powers of (-1) are incorrect. It should be k(E_P \cup < x,y > ) + |E_P| +1 in the numerator, and k(E_P) + |E_P| in the denominator.
• Page 262, Proof of Thm 9.15(a): The polynomials are in q and y = e^{-\lambda q}, and the proof is an easy piece of analysis. Just multiply up and write as \sum_{r,s} a_{r,s} q^r y^s = 0 where a_{r,s} \ne 0. By considering the limit q \to \infty, we find that a_{R,S} = 0, a contradiction, where S=\min s, and R is the maximum value of r corresponding to this value S.
• Page 264: Corollary 9.32 is sometimes referred to as Theorem 9.32.
• Page 274, line 17: The first term on the (first) RHS should be J M_\Lambda dM_\Lambda/d(\beta h).
• Page 275, line -5: Replace random-cluster by random-current.
• Page 353: Reference [11] was published by World Scientific, Singapore.
• Page 354: Reference [19] appeared in 2004.
• Page 366: Reference [232] concerns `transitions'.
• Page 369: The pages of [267] are 2089-2116.