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A finite lattice [tex]X[/tex] with so constructed boundary condition that [tex]M_s(X;T)\neq 0[/tex]

boundary condition - all spins in the boundary are up, in Ising model [tex]S_i=1, \forall i \in \partial X[/tex]

Wall - line that separates + and - sites.

Two probabilities

1) [tex]\omega_i(T)[/tex] - probability that at temperature [tex]T[/tex] site [tex]i[/tex] is occupied by spin -

2) [tex]W_{\Gamma}[/tex] - probability that at temperature [tex]T[/tex] polygon [tex]\Gamma[/tex] exists.

Can you tell me exactly what they suppose by polygon? There is a picture in page 247. How many polygons is on this picture?

Also can you explain me estimation (6.54) on page 248.