The “Mexican wave” came to worldwide prominence during the 1986 World Cup of football held in Mexico. A Mexican wave can propagate around a stadium as successive spectators stand, cheer, and raise their arms before sitting back down. Suppose we represent the seats in a stadium by a 2D lattice with periodic boundary conditions. Let the probability of a seat being occupied by a spectator be p and consider von-Neumann neighborhoods.

a). If q is the property that the Mexican wave can percolate around the stadium derive a difference equation q2 = f(q1) describing how the property at the scale of a single cell evolves to the scale of 2×2 blocks. Hence, find qs+1 = F[qs] for all s.

(b) Find the fixed points of the above difference equation and deter- mine a critical percolation threshold pc.

(c) The capacity of Optus Stadium is approximately 60,000. The average crowd of the Australian Rules Football teams Fremantle Dockers and West Coast Eagles are approximately 40,000 and 53,000 respectively. Without casting aspersions on the type of fans or team game style can a Mexican wave be sustained by each fan base?

(d) In COVID times the average attendance has dropped to around 22, 000 can a Mexican wave be sustained in these circumstances?