# GUE check of Singh Lemma 7 reduction: given gap s (pts +-s/2), P3(s,y)=int_{s/2<|x|<y} rho3/rho2 dx
# (expected # third points within y of midpoint). Average over s<eps with density ~ s^2.
from mpmath import mp,mpf,sin,pi,quad,matrix,det
mp.dps=30
S=lambda d: mpf(1) if d==0 else sin(pi*d)/(pi*d)
def rho(p): return det(matrix([[S(a-b) for b in p] for a in p]))
def P3(s,y):
    r2=rho([-s/2,s/2]); return quad(lambda x:2*rho([-s/2,s/2,x])/r2,[s/2,y])
for eps in [mpf('0.2'),mpf('0.1'),mpf('0.05'),mpf('0.025')]:
    y=eps**mpf('0.5')
    # E over s~3s^2/eps^3 on (0,eps)
    n=10; E=sum(3*((k+mpf(0.5))*eps/n)**2/eps**3*P3((k+mpf(0.5))*eps/n,y)*eps/n for k in range(n))
    print(float(eps),float(y),'E[N_y|s<eps]=',float(E),' /eps^2=',float(E/eps**2),' /eps^2.5=',float(E/eps**2.5))