FIG. 3: The average particle number and the condensate susceptibility as a function of µ near the transition between N = 0 and N = 1 at L = 2 and β = 0 . 43 for different values of L$_{t}$ . The solid lines show the fit of data to the effective quantum mechanics description discussed in the text. 0 0.2 0.4 0.6 0.8 1 < N > L$_{t}$=40 L$_{t}$=100 L$_{t}$=150 L$_{t}$=200 0.775 0.8 0.825 0.85 0.875 0 50 100 150 200 250 L$_{t}$=100 L$_{t}$=150 L$_{t}$=200 0 0.5 1 1.5 0 1 2 3 4 < N > L=2 L=4 L=6 L=8 L=12 L=16 0 0.5 1 1.5 0 1 2 3 4 < N > L=2 L=4 L=6 L=8 L=12 L=16 FIG. 4: The average particle number as a function of µ for at β = 0 . 43 (left) β = 0 . 50 (right) for different values of L . When β = 0 . 43 the data shown is for L$_{t}$ = 100 at L = 2 − − 8 , L$_{t}$ = 200 at L = 12 and L$_{t}$ = 300 at L = 16 . When β = 0 . 50 the data shown is for L$_{t}$ = 100 at L = 2 − − 6 , L$_{t}$ = 160 at L = 8 , L$_{t}$ = 200 at L = 12 and L$_{t}$ = 300 at L = 16 . 17