Consider a population consisting of individuals that are either of opinion A or opinion B. Initially the entire population is of opinion A. New information arrives, which is scrutinised by everyone and after doing this, each individual changes their opinion from A to B with probability p. Each individual has qc neighbours.

(a) Let P_∞ denote the probability that a randomly chosen person belongs to an infinite (system-spanning) cluster of B individuals. Determine in mean field the value pc of p at which P_∞ becomes non-zero.
(b) Determine the mean-field exponent, β, describing the behaviour of P_∞ ∝ (p−pc)β in the critical regime above p_c.
Now assume that the population can be considered to live on a one-dimensional square lattice of infinite extent.
(c) Determine the probability, P(s), that a randomly chosen individual belongs to a B-cluster of size s.
(d) Determine the average size of clusters of B peoplewhen visiting clusters by choosing a person at random as done in (c).

(e) Determine the exponent γ which describes the behaviour of the average size of B-clusters as the critical value of p is approached.