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

Corollary 6 ([Growth of the limiting spectral rates]). For GOE-almost every g , and every b with b+g1>0 , there are finite positive constants cg,b ,Cg,b such that cg,b (1+i)2/3 ≤gi+b≤Cg,b (1+i)2/3 (i≥1). They can be chosen …

Lemma 5 ([Spherical projection and pair tails]). Fix a field radius and a fixed number of replicas. On the spectral restrictions above, for every p with kp≥K and p≤C1rn , both the uncapped zero-field spherical law and the capped …

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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.

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