The topological charge distribution P(Q) is calculated for lattice ${rm CP}^{N-1}$ models. In order to suppress lattice cut-off effects we employ a fixed point (FP) action. Through transformation of P(Q) we calculate the free energy $F(theta)$ as a function of the $theta$ parameter. For N=4, scaling behavior is observed for P(Q), $F(theta)$ as well as the correlation lengths $xi(Q)$. For N=2, however, scaling behavior is not observed as expected. For comparison, we also make a calculation for the ${rm CP}^{3}$ model with standard action. We furthermore pay special attention to the behavior of P(Q) in order to investigate the dynamics of instantons. For that purpose, we carefully look at behavior of $gamma_{it eff}$, which is an effective power of P(Q)($sim exp(-CQ^{gamma_{it eff}})$), and reflects the local behavior of P(Q) as a function of Q. We study $gamma_{it eff}$ for two cases, one of which is the dilute gas approximation based on the Poisson distribution of instantons and the other is the Debye-Huckel approximation of instanton quarks. In both cases we find similar behavior to the one observed in numerical simulations.
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