Lemma 7 ([Two-replica overlap cells]). Let R_n= -1+2j/n:0≤j≤n and I_r=[r-1/n,r+1/n] . The overlap of two independent uniform cube spins has mass p_n(r)=2^-n n n(1+r)/2 , r∈ R_n, whereas the overlap of two independent uniform spherical …

We derive this form for cells centered at lattice points from the two-replica calculations in and . Set I(r)=12((1+r)(1+r)+(1-r)(1-r)), J(r)=-12(1-r^2). Uniform Stirling bounds on |r|≤ρ_0 put p_n(r) between fixed multiples of n^ -1/2 e^ -nI(r) . The same bounds for the gamma ratio, and the bounded logarithmic oscillation of f_n on I_r , put its cell integral between fixed multiples of n^ -1/2 e^ -nJ(r) . Since J(r)-I(r)=O(r^4) , this proves . The assumed oscillation of the positive angular weight gives by integration. The final assertion follows directly from the cell endpoints.