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In this work I present a numerical study of the Finite Size Scaling (FSS) ofa correlation length in the framework of the $CP ^{N-1}$ model by means of the1/N expansion. This study has been thought as propedeutical to the applicationof FSS to the measure on the lattice of a new coupling constant $f_{x}(1/R)$,defined in terms or rectangular Wilson Loops. I give also a perturbativeexpansion of $f_{x}(1/R)$ in powers of the corresponding coupling constant inthe $\overline{MS}$ scheme together with some preliminary numerical resultsobtained from the Polyakov ratio and I point out the conceptual problems thatlimit this approach.Comment: 17 pages in REVTex + 6 figures, to be published in International  Journal of Modern Physics

FINITE SIZE SCALING AND RUNNING COUPLING CONSTANT IN CPN-1 MODELS

FINITE SIZE SCALING AND RUNNING COUPLING CONSTANT IN CP^N-1 MODELS