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We study the dressing of operators and flows of corresponding couplings inmodels of {\it embedded} random surfaces. We show that these dressings can beobtained by applying the methods of David and Distler and Kawai. We considertwo extreme limits. In the first limit the string tension is large and thedynamics is dominated by the Nambu-Goto term. We analyze this theory around aclassical solution in the situation where the length scale of the solution islarge compared to the length scale set by the string tension. Couplings getdressed by the liouville mode (which is now a composite field) in a non-trivialfashion. However this does {\it not} imply that the excitations around aphysical ``long string" have a phase space corresponding to an extra dimension.In the second limit the string tension is small and the dynamics is governed bythe extrinsic curvature term. We show, perturbatively, that in this theory therelationship between the induced metric and the worldsheet metric is``renormalized", while the extrinsic curvature term receives a non-trivialdressing as well. This has the consequence that in a generic situation thedependence of couplings on the physical scale is different from that predictedby their beta functions.Comment: 14 pages. Preprint TIFR-TH-92/4

RENORMALIZATION OF COUPLINGS IN EMBEDDED RANDOM SURFACES