The energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other. For a droplet with perimeter L, the area is somewhere between (L − 2)/2 (the straight line) and (L /4) (the square box). The probability cost for introducing a droplet has the factor e, but this contributes to the partition function multiplied by the total number of droplets with perimeter L, which is less than the total number of paths of length L : N ( L ) < 4 2 L . So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by ∑ ∑ L L 2 4 2 L e − − 4 β β L ,