21 FIG. 4: A plot of log$_{10}$ ( f$_{k}$ ) versus k for the case of N = 1000, Q = 5 for 50 different configurations. Notice how for large k the magnitudes cover various decades. FIG. 5: The radius R$_{0}$ , r ( r $^{)}$, r ( s ) of the hole relative to the radius of the sphere at Q = 0, evaluated at different values of k before taking the limit . We see that the values of R$_{0}$ are lower than r ( r ) and show faster convergence, while r ( s $^{)}$, the naive definition of the radius converges towards zero and is dominated by fluctuations for large k . Now, let us prove that the inequalities R$_{0}$$_{k}$ < r ( r ) k $_{+1}$. Notice that configuration by configu- ration we have the following equality crosses the one defined by the fractions, even though configuration by configuration the naive definition gives a higher value. This is because the averaging procedure is different. Notice also how for low values of k the naive definition and the root definition agree for the most part. This is exactly what is expected when fluctuations are small. f$_{k}$ f$_{k}$$_{+1}$ = r − 2 k k r − 2( k +1) k +1 = ( r$_{k}$ r$_{k}$$_{+1}$ ) − 2 k r 2 k +1 (21) 2 4 6 8 10 12 14 k 5 10 15 20 log f_k ( ( * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 2 4 6 8 10 12 14 0.0 0.1 0.2 0.3 0.4 0.5 0.6 * Naive * Root * Fraction